MAD 2104

True/False: If A, B, and C are arbitrary sets with bijective functions f: A ⟶ B and g: B ⟶ C, then (g ∘ f) is also bijective.

True

True/False: If the square of every integer is negative, then 1 is negative

True

True/False: The function f : ℤ+ ⟶ ℝ defined by f(x) = 1/x is injective (one-to-one).

True

Complete this truth table for the expression ¬ 𝑝 . p | ¬ p F | _____ T | _____

True False

Which implication correctly expresses the meaning of the statement, "p is sufficient for q" ? 𝑞 → 𝑝 -or- 𝑝 → 𝑞

𝑝 → 𝑞

Let set A = { a, b, c } and set Z = { x, y, z }. Using these sets, answer the following question: Specify one element of the set Z x Z.

(x, x) (x, y) (x, z) (y, x) (y, y) (y, z) (z, x) (z, y) (z, z)

What is the least integer n such that the function 4x^3 + 2x^2 + x is O(x^n)?

3

When proving Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0, by induction, which statement correctly expresses the induction hypothesis? - Assume Σ(i=1, to k) of (2i + 1)^2=((k + 1)(2k + 1)(2k + 3))/3, for some integer k ≥ 0 - Assume Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3, for some integer k ≥ 0 - Assume Σ(i=1, to k) of (2i + 1)^2=((k + 1)(2k + 1)(2k + 3))/3, for some integer k ≥ 0 - Assume Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3, for all integers k ≥ 0

- Assume Σ(i=1, to k) of (2i + 1)^2=((k + 1)(2k + 1)(2k + 3))/3, for some integer k ≥ 0

Indicate the result of the bit string operation shown: ¬ 10001101

01110010

Given the Boolean matrices M and N as shown here, what is the matrix that equals M ∧ N ? M = 1 0 0 1 1 1 N= 1 1 0 0 0 1

1 0 0 0 0 1

Given the Boolean matrices M and N as shown here, what is the matrix that equals M ∨ N ? M = 1 0 0 1 1 1 N= 1 1 0 0 0 1

1 1 0 1 1 1

Indicate which of these functions is bijective (one-to-one and onto).There may be more than one or none. 'True' if the function is bijective; otherwise 'False'. 1) T/F: The function f : ℕ ⟶ ℝ defined by f(x) = 1/x is bijective. 2) T/F: The function f : ℕ ⟶ ℤ defined by f(x) = x is bijective. 3) T/F: The function f : ℤ ⟶ ℤ defined by f(x) = ⌊ x + 1 ⌋ is bijective. (Note the use of the floor function) 4) T/F: The function f : ℤ ⟶ {0, 1} defined by f(x) = 𝓧A(𝑥), the characteristic function of set A, where set A = {2n|n ∈ ℤ}, is bijective.

1) False 2) False 3) True 4) False

Consider proving the following statement using a direct proof. "If x is a non-zero rational number, then 1/x is rational." 1. What do you assume as true to begin the proof? 2. What do you demonstrate must be true to complete the proof? Answer 1: Assume x is a rational number and x is not zero. -OR- Assume x and 1/x are rational numbers. Answer 2: Show 1/x is a rational number when x is a non-zero rational number. -OR- Show 1/x is a non-zero rational number when x is a rational number..

1. Assume x is a rational number and x is not zero. 2. Show 1/x is a rational number when x is a non-zero rational number.

Identify the expressions that are logically equivalent to ¬ 𝑞 → ¬ 𝑝. There may be more than one or none. Select 'True' for each expression that is equivalent to the given expression; otherwise select 'False'. 1. ¬ 𝑞 → 𝑝 2. ¬ 𝑞 ∧ 𝑝 3. 𝑞 ∨ ¬ 𝑝 4. 𝑞 → 𝑝 5. ¬ 𝑝 → ¬ 𝑞 6. ¬ 𝑝 ∧ 𝑞 7. 𝑝 ∨ ¬ 𝑞 8. 𝑝 → 𝑞

1. False 2. False 3. True 4. False 5. False 6. False 7. False 8. True

Let P (𝑚, 𝑛) be a predicate defined for all 𝑚, 𝑛 ∈ ℤ, as "𝑚 + 𝑛 = 0". Indicate whether each of these statements using predicate P is true or false. 1. ∀𝑛 P ( 0, 𝑛 ) 2. ∀𝑚 P ( 𝑚, -2 ) 3. ∃!𝑚 P ( 𝑚, 2 ) 4. ∀𝑚 ∃𝑛 P ( 𝑚, 𝑛 ) 5. ∀x [ P ( x, 1 ) → 3x ≥ 0 ] 6. ¬ [ ∀𝑚 ∃𝑛 P ( 𝑚, 𝑛 ) ] ⟺ ¬ [ ∀𝑛 ∃𝑚 P ( 𝑚, 𝑛 )

1. False 2. False 3. True 4. True 5. False 6. True

Identify the expressions that are logically equivalent to ¬ ( 𝑝 ∨ ¬ 𝑞 ). There may be more than one or none. Select 'True' for each expression that is equivalent to the given expression; otherwise select 'False 1. 𝑞 → ¬ 𝑝 2. 𝑞 ∧ ¬ 𝑝 3. ¬ 𝑞 ∨ ¬ 𝑝 4. 𝑞 ∧ 𝑝 5. ¬ 𝑝 ∧ 𝑞 6. 𝑝 → 𝑞 7. ¬ 𝑝 ∨ 𝑞 8. ¬ 𝑝 → 𝑞

1. False 2. True 3. False 4. False 5. True 6. False 7. False 8. False

Indicate the result of the bit string operation shown. 10001101 ∧ 10001011

10001001

The binary expansion of the decimal number 206 is ___________(base2) is?

11001110(base2) 206-128=78 78-64=14 14-8=6 6-4=2 2-2=0 2^7 | 2^6 | 2^5 | 2^4 | 2^3 | 2^2 | 2^1 | 2^0 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 1 | 1 | 0 | 0 | 1 | 1 | 1 | 0

What is 𝝓 (21), where 𝝓 is the Euler-𝝓 function?

12

The octal expansion of the decimal number 105 is ___________eight.

151

Know how to complete membership table | A | B | C | (B - C) | (A - B) | A - (B - C) | (A - B) - C | 0 | 0 | 1 | 1) ? | 2) ? | 3)? | 4) ? | 0 | 1 | 0 | 5) ? | 6) ? | 7)? | 8) ? | 0 | 1 | 1 | 9) ? | 10) ? | 11)? | 12) ? | 1 | 0 | 0 | 13) ? | 14) ? | 15)? | 16) ? | 1 | 0 | 1 | 17) ? | 18) ? | 19)? | 20) ? | 1 | 1 | 0 | 21) ? | 22) ? | 24)? | 24) ? | 1 | 1 | 1 | 25) ? | 26) ? | 27)? | 28) ?

1: 0 2: 0 3: 0 4: 0 5: 1 6: 0 7: 0 8: 0 9: 0 10: 0 11: 0 12: 0 13: 0 14: 1 15: 1 16: 1 17: 0 18: 1 19: 1 20: 0 21: 1 22: 0 23: 0 24: 0 25: 0 26: 0 27: 1 28: 0

Consider proving the following statement using a proof by cases. "For all positive integers n ≤ 3, 2n ≤ n+3." 1: What 3 cases do you use for this proof: Case 1: n = ?; Case 2: n = ?; Case 3: n = ? What do you demonstrate must be true to complete the proof of each case? .

1: Case 1: n = 1; Case 2: n = 2; Case 3: n = 3. 2: Show 2n ≤ n+3 when n is a positive integer ≤ 3.

Indicate which of the following areas of computer science often utilize modular arithmetic. Select 'True' if modular arithmetic is used often; otherwise select 'False'. There may be more than one or none. 1: Representing bit strings and bit string operations - AND, OR, XOR, NOT 2: Computing matrix operations - meet, join, Boolean product 3: Hashing to reduce an extremely large range of values to a limited range of integers 4: Generating pseudo-random number sequences for randomizing program choices 5: Evaluating algorithm timing functions for efficiency 6: Data encryption and decryption, such as the RSA public-key cryptosystem

1: False 2: False 3: True 4: True 5. False 6. True

Indicate which of the following can be used to complete this sentence: For arbitrary positive integers a, b, and c, with a ≠ 0 and b ≠ 0, if a | b and a | c, then ______________________. Select 'True' if the phrase is a correct completion of the statement; otherwise select 'False'. There may be more than one or none. 1: b | c 2: a | (b c) 3: a | (b +c) 4: b | (a c)

1: False 2: True 3: True 4: False

Consider the following statement: "The function f : ℝ ⟶ ℤ defined by f(x) = ⌊ 2x ⌋ is surjective (onto)." 1: What must you demonstrate to prove the statement? 2: What must you demonstrate to disprove the statement?

1: Show every element y in the co-domain of function f is the image of some element of the domain of function f. 2: Show some element y in the co-domain of function f is the image of no element of the domain of function f.

ndicate which of the following can be used to complete this sentence: For arbitrary positive integers a, b, c, and m with m>1, if a ≡ b (mod m) and c ≡ d (mod m), then ___________________________. Select 'True' if the phrase is a correct completion of the statement; otherwise select 'False'. There may be more than one or none. 1: (a + c) ≡ (b + d) (mod m) 2: (a c) ≡ (b d) (mod m) 3: a (mod m) = c (mod m) 4: b (mod m) = d (mod m)

1: True 2: True 3: False 4: False

With the definition for adult as "a person age 18 or older" and the universe of discourse for the variable x as "adults living in Florida," we define the predicate, R(x) as "person x is registered to vote." Using these definitions, answer the following questions about this English sentence: "There is some adult living in Florida who is registered to vote." 1. Indicate which quantified symbolic expression accurately represents the given English sentence. 2. Indicate which quantified symbolic expression represents the negation of the given English sentence. 3. Indicate which statement accurately expresses the negation of the given English sentence.

1: ∃ x R( x ) 2: ∀ x ¬ R( x ) 3: Every adult living in Florida is not registered to vote.

Let A = { a, b, c } and set B = { a, b, { b }, c, d }. Using these sets, answer the following: What is | A |, the cardinality of set A?

3

Let set A = { a, b, c } and set Z = { x, y, z }. Using these sets, answer the following question: How many subsets of set A have cardinality 2?

3

The decimal expansion of the hexadecimal number 4B(base16) is _______________(base ten).

75

Consider the following problem: 7^92(mod 11) = _________ Show how Fermat's Little Theorem can be used to solve this problem. Express your answer as a non-negative integer less than the modulus.

7^92(mod11)=8 calculation: 7^10=1(mod11) =7^10*9+2 =(7^10)9 + 7^2 =1^9 + 49 7^92(mod11)=8

Let set A = { a, b, c } and set Z = { x, y, z }. Using these sets, answer the following question: What is the cardinality of the set Z x Z, the cross product of set Z with itself?

9

Indicate a reason for each assertion in the argument below. Choose your answers from the given list of Rules of Inference. An item from the list may be used as a reason more than once or not at all. Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Assertion | Reason Premise 1. ( s ˄ ¬ r ) → w | Given Premise 2: ( ¬ s → ¬ t ) ˅ r | Given Premise 3: t ˄ ¬ r | Given A. ¬ r | ____________(with premise 3) B. ( ¬ s → ¬ t ) | ____________(with step A and premise 2) C. t | _____________(with premise 3) D. s | ______________(with steps B, C) E. s ˄ ¬ r | ______________(with steps A, D) F. w | ______________(with step E and premise 1)I

A. Conjunction elimination B. Disjunctive syllogism C. Conjunction elimination D. Modus tollens E. Conjunction introduction F. Modus ponens

Let the function f : ℕ → ℝ be defined recursively as follows: Initial Condition: f (0) = 1 Recursive Part: f (n + 1) = 3 * f (n), for n ≥ 0 Consider how to prove the following statement about this given function f using induction. f (n) = 3^n, for all nonnegative integers n. Basis step: For n = 0, f(n) = f(0) = 1; also 3^n = 3^0 = 1, so f(n) = 3^n for n = 0. Inductive Hypothesis: Assume f(k) = 3^k for some integer k ≥ 0. Which is a correct way to complete the Inductive Step for this proof? A. When the inductive hypothesis is true, f(k+1) = 3*f(k) = 3*3^k = 3^(k+1). B. f(k+1) = 3*f(k), which confirms the recursive part of the definition. C. When f(k+1) = 3^(k+1) = 3*3^k; also f(k+1) = 3*f(k), so f(k) = 3^k, confirming the induction hypothesis. D. When the inductive hypothesis is true, f(k+1) = 3^(k+1) = 3*3^k = 3*f(k), which confirms the recursive part of the definition.

A. When the inductive hypothesis is true, f(k+1) = 3*f(k) = 3*3^k = 3^(k+1).

The membership table in the previous problem shows that the two given sets, A - (B - C) and (A - B) - C, ARE or ARE NOT equal.

ARE NOT

Consider this congruence: 2𝑥 ≡ 4 (mod 5) Which statement correctly identifies an inverse needed to solve the given congruence? - An inverse of 2 (mod 5) is 1 since GCD(2, 5) is 1. - An inverse of 2 (mod 5) is 3 since 3*2 = 6 = 1 (mod 5) - An inverse of 4 (mod 5) is 4 since 4*4 = 16 = 1 (mod 5) - An inverse of 2 (mod 5) does not exist since GCD(2, 4) is not 1.

An inverse of 2 (mod 5) is 3 since 3*2 = 6 = 1 (mod 5)

Consider the following statement: "The function f : ℝ ⟶ ℤ defined by f(x) = ⌊ 2x ⌋ is injective (one-to-one)." 1: What must you demonstrate to prove the statement? 2: What must you demonstrate to disprove the statement?

Answer 1: Show that for all elements x and y in the domain of function f, f(x) = f(y) implies x = y. Answer 2: Show that there are elements x and y in the domain of function f, such that f(x) = f(y) but x ≠ y.

Which of these given arguments uses no fallacy and therefore is a valid argument? Argument A: Proving: For every real number x, x < x + 1. 1. Let x be an arbitrary real number. 2. We know that 0 < 1. 3. Adding x to both sides, gives x + 0 < x + 1. 4. And that gives the equivalent inequality x < x + 1. 5. So for every real number x, x < x + 1. Argument B: Proving: For integers x and y, if xy is a multiple of 5, then x is a multiple of 5 and y is a multiple of 5. 1. Let x and y be integers with xy a multiple of 5. 2. x is a multiple of 5 means x = 5k, for some integer k. 3. Similarly, y is a multiple of 5 means y = 5j for some integer j. 4. Substituting for x and y, we get xy = (5k)(5j) = 5(5kj). 5. Since 5kj is an integer, the product xy, which equals 5(5kj), is a multiple of 5. 6. So xy is a multiple of 5, when x is a multiple of 5 and y is a multiple of 5. Argument C: Proving: For every positive real number x, x + 1/x ≥ 2. 1. Let x be a positive real number with x + 1/x ≥ 2. 2. Multiplying both sides by x, we have x^2 + 1 ≥ 2x. 3. So by algebra, we get x^2 - 2x + 1 ≥ 0, or (x-1)2 ≥ 0. 4 Since it is true that the square of any real number is positive, 5. (x-1)^2 ≥ 0 confirms that x + 1/x ≥ 2, for every positive real number x. Argument D: Proving: For all integers m and n, if m and n are odd, then (m+n) is odd. 1. Let m and n be integers. 2. We know that when m and n are even, then (m+n) is even. 3. So if m and n are odd, (m+n) is odd.

Argument A

Complete the argument below that shows the following logical expression is a tautology: ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 Indicate which logical equivalence is the reason for each step. Assertion | Reason ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 | Given expression ⇔ ¬ (( 𝑞 ∨ 𝑝 ) ∧ ¬ 𝑞) ∨ 𝑝 | Implication ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ¬¬ 𝑞) ∨ 𝑝 | DeMorgan's ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ 𝑞) ∨ 𝑝 | Double negation ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ( 𝑞 ∨ 𝑝 ) | ___________________________ "Implication", "Identity", "DeMorgan's", "Contrapositive", "Commutative", "Double negation", "Tautology", "Distributive", "Associative"

Associative

Consider proving the following statement by proving the contrapositive. "For all integers m and n, if (m n) is even, then m and n are even" What do you assume as true to begin the proof? Assume m and n are integers Assume m and n are odd integers and (m n) is even. Assume m and n are even integers. Assume m and n are even integers and (m n) is odd. Assume m and n are odd integers.

Assume m and n are odd integers.

Let the function f : ℕ → ℝ be defined recursively as follows: Initial Condition: f (0) = 1 Recursive Part: f (n + 1) = 3 * f (n), for n ≥ 0 Consider how to prove the following statement about this given function f using induction. f (n) = 3^n, for all nonnegative integers n. Which is a correct way to prove the Basis Step for this proof? A. For n = 1, f(n) = f(1) = 3*f(0) = 3; also 3^n= 3^1 = 3, so f(n) = 3^n for n = 1. B. For n = 0, f(n) = f(0) = 1; also 3^n = 3^0 = 1, so f(n) = 3^n for n = 0. C. For n = k+1, f(k+1) = 3^(k+1) when f(k) = 3^k for some integer k ≥ 0, so f(n) = 3^n for n = k+1. D. For n = k, assume f(k) = 3^k for some integer k ≥ 0, so f(n) = 3^n for n = k.

B. For n = 0, f(n) = f(0) = 1; also 3^n = 3^0 = 1, so f(n) = 3^n for n = 0.

Complete this proof by identifying the statement that correctly matches each step in the inductive proof of this assertion, "For all integers n ≥ 4, 2n < 2^n." Basis step: Inductive Hypothesis: Inductive Step: Conclusion: - When k ≥ 4 and (2k) < (2^k), 2(k+1) = 2k+2 < 2k+2k = 2(2k) = 2^(k+1), so 2(k+1) < 2^(k+1). - When n = 4, (2n) = 8 and (2^n) = 16, so (2n) < (2^n) for n = 4. - Assume that (2k) < (2^k) for all integers k ≥ 4. - For all integers n, (2n) < (2^n). - For all integers n ≥ 4, (2n) < (2^n). - Assume that (2k) < (2^k) for all integers k ≥ 4. - Assume that 2(k+1) < 2^(k+1), when (2k) < (2^k). - Assume that for some integer k ≥ 4, 2k < (2^k). - Prove that (2k) < (2^k), for some integer k≥ 4. - When n = 1, (2n) = 2 and (2^n) = 2, so (2n) ≤ (2^n) for n = 1. - When (2k) < (2^k) for all integers k ≥ 4, 2(k+1) < 2^(k+1), since (k+1) ≥ 4.

Basis Step: When n = 4, (2n) = 8 and (2^n) = 16, so (2n) < (2^n) for n = 4. Inductive Hypothesis: Assume that for some integer k ≥ 4, 2k < (2^k). Inductive Step: When k ≥ 4 and (2k) < (2^k), 2(k+1) = 2k+2 < 2k+2k = 2(2k) = 2^(k+1), so 2(k+1) < 2^(k+1). Conclusion: For all integers n ≥ 4, (2n) < (2^n).

Let A = { a, b, c } and set B = { a, b, { b }, c, d }. Using these sets, answer the following: The empty set, ∅. is a subset of which of these sets A and B? -only set B -both sets A and B -only set A -neither set A nor set B

Both set A and B

Let the function f : ℕ → ℝ be defined recursively as follows: Initial Condition: f (0) = 1 Recursive Part: f (n + 1) = 3 * f (n), for n ≥ 0 Consider how to prove the following statement about this given function f using induction. f (n) = 3^n, for all nonnegative integers n. Basis step: For n = 0, f(n) = f(0) = 1; also 3^n = 3^0 = 1, so f(n) = 3^n for n = 0. Which is a correct way to state the Inductive Hypothesis for this proof? A. Prove f(k) = 3^k for some integer k ≥ 0. B. Prove f(k) = 3^k for all integers k ≥ 0. C. Assume f(k) = 3^k for some integer k ≥ 0. D. Assume f(k+1) = 3^(k+1) when f(k) = 3^k for some integer k ≥ 0.

C. Assume f(k) = 3^k for some integer k ≥ 0.

Let the function f : ℕ → ℝ be defined recursively as follows: Initial Condition: f (0) = 1 Recursive Part: f (n + 1) = 3 * f (n), for n ≥ 0 Consider how to prove the following statement about this given function f using induction. f (n) = 3^n, for all nonnegative integers n. Basis step: For n = 0, f(n) = f(0) = 1; also 3^n = 3^0 = 1, so f(n) = 3^n for n = 0. Inductive Hypothesis: Assume f(k) = 3^k for some integer k ≥ 0. Inductive Step: When the inductive hypothesis is true, f(k+1) = 3*f(k) = 3*3^k = 3^(k+1). Which is a correct way to state the conclusion for this proof? A. By the principle of mathematical induction, f(k) = 3^k implies f(k+1) = 3^(k+1) for all integers k ≥ 0. B. By the principle of mathematical induction, f(k) = f(k+1) for all integers k ≥ 0. C. By the principle of mathematical induction, f(n+1) = 3*f(n) for all integers n ≥ 0. D. By the principle of mathematical induction, f(n) = 3^n for all integers n ≥ 0.

D. By the principle of mathematical induction, f(n) = 3^n for all integers n ≥ 0.

Complete the argument below that shows the following logical expression is a tautology: ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 Indicate which logical equivalence is the reason for each step. Assertion | Reason ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 | Given expression ⇔ ¬ (( 𝑞 ∨ 𝑝 ) ∧ ¬ 𝑞) ∨ 𝑝 | Implication ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ¬¬ 𝑞) ∨ 𝑝 | ____________________________ [ Select ] ["Double negation", "Commutative", "Implication", "DeMorgan's", "Identity", "Associative", "Tautology", "Contrapositive", "Distributive"]

DeMorgan's

Complete the argument below that shows the following logical expression is a tautology: ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 Indicate which logical equivalence is the reason for each step. Assertion | Reason ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 | Given expression ⇔ ¬ (( 𝑞 ∨ 𝑝 ) ∧ ¬ 𝑞) ∨ 𝑝 | Implication ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ¬¬ 𝑞) ∨ 𝑝 | DeMorgan's ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ 𝑞) ∨ 𝑝 | ___________________ "Implication", "Distributive", "Commutative", "Associative", "Double negation", "DeMorgan's", "Tautology", "Identity", "Contrapositive"

Double Negation

Let A = { a, b, c } and set B = { a, b, { b }, c, d }. Using these sets, answer the following: True or False? B ⊆ A

False

Let set A = { a, b, c } and set Z = { x, y, z }. Using these sets, answer the following question: True or False? Z ⊆ Z x Z

False

T/F: For arbitrary predicates P and Q, ∀x [P (x) ∨ Q (x)] ⟹ ∀x P (x) ∨ ∀x Q (x).

False

True/False For all sets A and B, if A ⋃ B = A, then A ⊂ B.

False

True/False If n is an arbitrary composite integer, then n has a factor less than or equal to 1/n.

False

True/False: For all sets A, B, and C, A ⋃ (B ⋂ C) = (A ⋂ B) ⋃ (A ⋂ C).

False

True/False: For all sets A, B, and C, if A ⋃ C = B ⋃ C, then A = B.

False

True/False: For arbitrary positive integers a, b, and c, with a ≠ 0, if a | (b + c), then a | b and a | c.

False

True/False: If A = { a, b, c } and B = { b, { c }}, then | 𝓟 (A × B) | = 32.

False

True/False: If a and b are integers and p is a prime number such that p | ab, then p | a and p | b.

False

True/False: If the product of two arbitrary integers x and y is even, then x is even and y is even.

False

True/False: The function f : ℝ ⟶ ℝ defined by f(x) = 2x^4 - 5x^3 + x^2 + 10x - 30 is O(x^3).

False

True/False: The function f : ℝ ⟶ ℤ defined by f(x) = x(mod 13) is a bijection.

False

True/False: The function f : ℤ ⟶ ℤ defined by f(x) = ⌊ x/2 ⌋ is a bijection. [Note the use of the floor function in the definition of function f.]

False

True/False: The function f : ℤ+ ⟶ {0, 1} defined by f(x) = 𝓧A(𝑥), the characteristic function of set A, where set A = {2n|n ∈ ℤ}, is injective (one-to-one).

False

Complete this truth table for the expression ( 𝑝 ⊕ 𝑞 ). That's the XOR operation. p | q | p ⊕ q F | F | _____________ F | T | _____________ T | F | _____________ T | T | _____________

False True True False

To disprove the following statement by counter-example, what do you need to demonstrate? "Every subset of ℤ, the set of integers, has a least element." - Identify a subset of integers that has a least element. - Identify a subset of integers that has no least element. - Identify 2 integers that bound an interval of real numbers with no l east element. - Select an arbitrary finite set of integers and identify the least element.

Identify a subset of integers that has no least element.

Given the following proposition definitions: p = "new software is available" q = "the computer is upgraded" Indicate which English sentence has equivalent meaning to the expression ¬𝑞 → ¬𝑝. - If new software is not available, the computer is not upgraded. - If the computer is not upgraded, then new software is not available. - If new software is available, the computer is upgraded. - If the computer is upgraded, then new software is available.

If the computer is not upgraded, then new software is not available.

Choose the English sentence that expresses the converse of this statement: "If new software is installed, the computer needs to be re-started." - If new software has not been installed, the computer does not need to be re-started. - If the computer does not need to be re-started, new software has not been installed. - If new software has not been installed, the computer needs to be re-started. - If new software has been installed, the computer does not need to be re-started. -OR- - If the computer needs to be re-started, new software has been installed.

If the computer needs to be re-started, new software has been installed.

Complete the argument below that shows the following logical expression is a tautology: ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 Indicate which logical equivalence is the reason for each step. Assertion | Reason ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 | Given expression ⇔ ¬ (( 𝑞 ∨ 𝑝 ) ∧ ¬ 𝑞) ∨ 𝑝 | ___________________________________________ "Distributive", "Contrapositive", "Implication", "Tautology", "Double negation", "Commutative", "DeMorgan's", "Associative", "Identity"

Implication

Consider proving the following statement using a proof by contradiction. "The sum of an odd integer and an even integer is odd." What do you assume as true to begin the proof? Let m and n be even integers with (m + n) an odd integer. Let m be an odd integer and n be an even integer with (m + n) even. Let m and n be odd integers with (m + n) also an odd integer. Let m be an odd integer and n be an even integer. Let m and n be integers with (m + n) an odd integer.

Let m be an odd integer and n be an even integer with (m + n) even.

Given the Boolean matrices M and N as shown here, what is M ⊙ N ? M = 1 0 0 1 1 1 N= 1 1 0 0 0 1

M ⊙ N is not defined because the dimension of matrix N should be k x j to appropriately match the dimension of matrix M (j x k).

Choose the English sentence that expresses the negation of this statement: "If new software is installed, the computer needs to be re-started." - New software is not installed and the computer does not need to be re-started. - New software is not installed and the computer needs to be re-started. - New software is installed and the computer needs to be re-started. - New software is installed and the computer does not need to be re-started.

New software is installed and the computer does not need to be re-started.

When selecting between two algorithms whose timing functions have different rates of growth, the algorithm with the timing function having the slowest growth rate is preferred. Which of these classes of function growth has the slowest growth rate? -O(2n) exponential growth -O(log n) logarithmic growth -O(n2) polynomial growth -O(n) linear growth

O(log n) logarithmic growth

When proving Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0, by induction, what would be proved in the basis step? - Prove the formula is true for some non-negative integer k - Prove the formula is true for n = 1 - Prove the formula is true for some non-negative integer (k + 1) - Prove the formula is true for n = 0

Prove the formula is true for n = 0

Consider proving the following statement by proving the contrapositive. "For all integers m and n, if (m n) is even, then m and n are even" What do you demonstrate must be true to complete the proof? Show (m n) must be odd when m and n are odd. Show (m n) must be odd when m and n are even. Show (m n) must be even when m and n are even. Show (m n) must be even when m and n are odd.

Show (m n) must be odd when m and n are odd.

Given a specific function f : ℝ ⟶ ℤ, which of the following approaches could best be used to prove function f IS injective (one-to-one)? - Show that for all elements a and b in the domain set ℝ, for f(a) and f(b) in the range of function f, f(a) = f(b) only when a = b. - Show that every element b in the co-domain set ℤ, has a corresponding element a in the domain set ℝ, such that function f maps a to b. - Show that there are elements a and b in the domain set ℝ, such that f(a) = f(b), but a ≠ b. - Show that there is some element b in the co-domain set ℤ that is not the image of any element of the domain set ℝ.

Show that for all elements a and b in the domain set ℝ, for f(a) and f(b) in the range of function f, f(a) = f(b) only when a = b.

Consider proving the following statement using a proof by contradiction. "The sum of an odd integer and an even integer is odd." What do you demonstrate must be true to complete the proof? Identify an even integer m and an odd integer n such that (m + n) is odd. Identify some specific odd integers m and n such that (m + n) is odd. Show that (m + n) must be even when m and n are both even. Show that for some specific proposition t, a contradiction, t ∧ ¬ t, is demonstrated.

Show that for some specific proposition t, a contradiction, t ∧ ¬ t, is demonstrated.

Complete the argument below that shows the following logical expression is a tautology: ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 Indicate which logical equivalence is the reason for each step. Assertion | Reason ((𝑝 ∨ 𝑞) ∧ ¬ 𝑞) → 𝑝 | Given expression ⇔ ¬ (( 𝑞 ∨ 𝑝 ) ∧ ¬ 𝑞) ∨ 𝑝 | Implication ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ¬¬ 𝑞) ∨ 𝑝 | DeMorgan's ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ 𝑞) ∨ 𝑝 | Double Negatoin ⇔ (¬ ( 𝑞 ∨ 𝑝 ) ∨ ( 𝑞 ∨ 𝑝 ) | Associative T | ______________________ "Identity", "Distributive", "DeMorgan's", "Double negation", "Implication", "Tautology", "Associative", "Commutative", "Contrapositive"

Tautology

True/False: For arbitrary positive integers a, b and m, with m = LCM(a, b), if c is a positive integer such that m | c, then a | c and b | c.

True

True/False: For arbitrary positive integers a, b, and m with m>1, if a = b + km, for some integer k, then a ≡ b (mod m).

True

True/False: If A = { a, b, c } and B = { b, { c }}, then Ø ∈ 𝓟 (A x B).

True

Are expressions ( ¬ 𝑞 → 𝑝 ) and ( ¬ 𝑝 ∨ 𝑞 ) logically equivalent, as determined by a truth table?

The two expressions ARE NOT logically equivalent.

Consider this problem that can be solved using the pigeonhole principle. Let the set S = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 }. Assume numbers are selected randomly (without looking at the list) from set S and that once a number is selected, it cannot be selected again (no duplicates). Determine the minimum number of integers, n, that must be selected from the set 𝑆 to guarantee that some pair of the n selected integers must have a sum of exactly 15. Which of the following best explains how the solution can be determined by framing the problem as equivalent to the generalized pigeonhole principle ? - The largest number (pigeon) is 14 and the smallest is 1, so 14-1 = 13 numbers must be selected to produce a pair of numbers that sum to 15. - There is 1 target result sum (pigeonhole) and 7 pairs that produce the desired sum (pigeons), so at most 7 numbers can be selected for a pair of numbers to sum to 15. - There are 14 numbers (pigeons) and 7 pairs that produce the given sum (pigeonholes), so at least 8 numbers must be selected to guarantee a pair of numbers that sum to 15. - Once a single number (pigeon) has been selected, there is only one other number (pigeon) that will match it to give the required sum, so exactly 2 numbers will be selected to give a pair of numbers that sum to 15.

There are 14 numbers (pigeons) and 7 pairs that produce the given sum (pigeonholes), so at least 8 numbers must be selected to guarantee a pair of numbers that sum to 15.

T/F: For arbitrary predicates P and Q, ∀x [P (x) ∧ Q (x)] ⟺ ∀x P (x) ∧ ∀x Q (x).

True

True/False For all sets A and B, ∅ ⊆ (A - B).

True

True/False: For an arbitrary function f, if f is a bijection, then its inverse is also a bijection.

True

True/False: For arbitrary integers a, b, and d, with d ≠ 0, if d = GCD(a, b) then there are integers s and t such that d = as + bt.

True

True/False: For arbitrary positive integers a and b, if 1 is a linear combination of a and b, then GCD(a, b) = 1.

True

Use the Euclidean algorithm to determine the GCD(308, 124). Show your work. Then express the GCD(308, 124) value you identify as a linear combination of 308 and 124. Show your work.

Your Answer: 308 = 2(124) + 60 --> 60 = 1(308) - 2(124) 124 = 2(60) + 4 --> 4 = 1(124) - 2(60) 60 = 4(15) + 0 GCD=4 4 = 1(124) -2[1(308 - 2(124)] = 1(124) - 2(308) + 4(124) = -2(308) + 5(124) is the linear combination

When proving Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0. by induction, which statement correctly expresses the final conclusion of the proof ? a) Prove Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0 b) Prove Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 1 c) Prove Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3, for some integer k ≥ 0 d) Prove Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3, for all integers k ≥ 0

a) Prove Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major c → s | Premise 2: Tom being smart is necessary for Tom to be a computer science major ¬ d → ¬ s | Premise 3: If Tom does not know discrete math Tom is not smart ¬ e | Premise 4: Tom is not an engineering major ____________________________________ | Steps 1, 4, Disjunctive syllogism s e c d

c

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major _____________________________________ | Premise 2: Tom being smart is necessary for Tom to be a computer science major s ∨ c s ∧ c s → c c → s

c → s

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason _________________________________ | Premise 1: Tom is a computer science major or an engineering major c ∨ e c → e e → c c ∧ e

c ∨ e

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major c → s | Premise 2: Tom being smart is necessary for Tom to be a computer science major ¬ d → ¬ s | Premise 3: If Tom does not know discrete math Tom is not smart ¬ e | Premise 4: Tom is not an engineering major c | Steps 1, 4, Disjunctive syllogism s | Steps 2, 5, Modus ponens ____________________________________ | Steps 3, 6, Modus tollens d c s e

d

When proving Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0, by induction, which statement correctly expresses what is proved in the induction step ? a) Assuming the induction hypothesis is true, prove: Σ(i=1, to k) of (2i + 1)^2=((k + 1)(2k + 1)(2k + 3))/3 b) Prove Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3, for some integer k ≥ 0 c) Prove Σ(i=1, to n) of (2i + 1)^2=((n + 1)(2n + 1)(2n + 3))/3, for all integers n ≥ 0 d)Assuming the induction hypothesis is true, prove Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3

d) Assuming the induction hypothesis is true, prove Σ(i=1, to k +1) of (2i + 1)^2=((k + 2)(2k + 3)(2k + 5))/3

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major c → s | Premise 2: Tom being smart is necessary for Tom to be a computer science major ¬ d → ¬ s | Premise 3: If Tom does not know discrete math Tom is not smart ¬ e | Premise 4: Tom is not an engineering major c | Steps 1, 4, Disjunctive syllogism s | Steps 2, 5, Modus ponens d | Steps 3, 6, Modus tollens Therefore, based on this argument above, we conclude that Tom ________________ _____________ __________.

knows discrete math.

To complete the division algorithm equation, a = mq + r, using a = - 56 and m = 5, which of the following gives appropriate values for integers q and r, with r expressed as a non-negative integer between 0 and (m-1), inclusive. - q = 12; r = - 4 - q = -11; r = -1 - q = -12; r = 4 - q = 11; r = 1

q = -12; r = 4

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major c → s | Premise 2: Tom being smart is necessary for Tom to be a computer science major ¬ d → ¬ s | Premise 3: If Tom does not know discrete math Tom is not smart ¬ e | Premise 4: Tom is not an engineering major c | Steps 1, 4, Disjunctive syllogism _____________________________________ | Steps 2, 5, Modus ponens c d s e

s

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. Which of these sets equals f^(-1)(T), the pre-image of set T under function f ? Set A Set S {0, 1} The empty set The positive integers

set A

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. What is the domain of this function f ? - Set S - Set A - Set T - Positive integers - {0, 1}

the positive integers

Define S, a set of bit strings, recursively as follows. Initial Condition: 0 ∈ S Recursion: If m ∈ S then m11 ∈ S. Which of the following best describes set S? - the set of all bit strings with odd length that start with a 0 but contain no other 0's - the set of all bit strings with odd length that begin with a 0 and end with a 1 - the set of all bit strings with odd length that begin with a 0 - the set of all bit strings that end with 11

the set of all bit strings with odd length that start with a 0 but contain no other 0's

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. Which of these sets equals f (S), the image of set S under function f ? - {0} - {1} - {0, 1} - Positive integers - The empty set

{ 0, 1 }

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. What is the co-domain of this function f ? - Set S - Set A - Set T - Positive integers - {0, 1}

{ 0, 1 }

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. Which of these sets equals the intersection of set T with the co-domain of function f ? - {1, 2, 3} - {0} - {1} - {0, 1} - The empty set

{ 1 }

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. Which of these sets correctly lists all the element of set T ? - {1, 2, 3} - {3, 4, 5, 6} - {0, 1} - Positive integers - The empty set

{ 1, 2, 3 }

Consider the function f : ℝ ⟶ ℤ defined by f(x) = ⌊ (x^2 + 5)/5 ⌋ Notice the use of the floor function in this definition of function f. What is the range of this function f ? -{ x | x ∈ ℤ } -{ x | x ∈ ℤ +} -{ 1, 2, 4, 6, 8, 10, 13, 17, 20 ... } -{ x | x ∈ ℝ }

{ 1, 2, 4, 6, 8, 10, 13, 17, 20 ... }

S, a subset of integers, is defined recursively below. Initial Condition: 2 ∈ S Recursive Step: If n ∈ S, then 3+n ∈ S. Which of the sets below is equal to S? { 2 + 3^n | n ∈ ℤ+} { 2 + 3^n | n ∈ ℕ } { 2 + 3n | n ∈ ℤ+ } { 2 + 3n | n ∈ ℕ}

{ 2 + 3n | n ∈ ℕ}

Consider the function f : ℤ+ ⟶ { 0, 1 } defined by f(x) = 𝓧(subA)(𝑥), the characteristic function of set A, where set A = { 5n | n ∈ ℤ }, the multiples of 5. Also, let set S = { x ∈ ℤ | 3 ≤ x ≤ 6 } and set T = { x ∈ ℤ | 1 ≤ x ≤ 3 }. Using these given definitions, answer the following questions. Which of these sets correctly lists all the elements of set S ? - {0, 1} - {0, 5, 10,...} - {1, 2, 3} - {3, 4, 5, 6} - {0, 1, 2,...}

{ 3, 4, 5, 6 }

Let A = { a, b, c } and set B = { a, b, { b }, c, d }. Using these sets, answer the following: What is A ⋂ B? -{a, c} -{a, {b}, c, d} -{a, b, {b}, c, d} -{a, b, c}

{ a, b, c }

Let A = { a, b, c } and set B = { a, b, { b }, c, d }. Using these sets, answer the following: What is A ⋃ B? -{a, c} -{a, {b}, c, d} -{a, b, {b}, c, d} -{a, b, c}

{ a, b, { b }, c, d }

Consider the function f : ℝ ⟶ ℤ defined by f(x) = ⌊ (x^2 + 5)/5 ⌋ Notice the use of the floor function in this definition of function f. What is the domain of this function f ? -{ x | x ∈ ℤ } -{ x | x ∈ ℤ +} -{ x | x ∈ N } -{ x | x ∈ ℝ }

{ x | x ∈ ℝ }

Consider the function f : ℝ ⟶ ℤ defined by f(x) = ⌊ (x^2 + 5)/5 ⌋ Notice the use of the floor function in this definition of function f. What is the co-domain of this function f ? -{ x | x ∈ ℤ } -{ x | x ∈ ℤ +} -{ x | x ∈ N } -{ x | x ∈ ℝ }

{ x | x ∈ ℤ }

Let A˅k = { x ∈ ℝ | 0 ≤ x ≤ 1/k }, for each positive integer k. What is ⋂(i=1 to n) of A˅k, where n is an arbitrary integer ≥ 0 ? - { x ∈ ℝ | 0 ≤ x ≤ n } - ∅ - { x ∈ ℝ | 0 ≤ x ≤ 1} - { x ∈ ℝ | 0 ≤ x ≤ 1/ n }

{ x ∈ ℝ | 0 ≤ x ≤ 1/ n }

Let A˅k = { x ∈ ℝ | 0 ≤ x ≤ 1/k }, for each positive integer k. What is ⋃(i=1 to n) of A˅k - { x ∈ ℝ | 0 ≤ x ≤ 1} - ∅ - { x ∈ ℝ | 0 ≤ x ≤ 1/3 } - { x ∈ ℝ | 0 ≤ x ≤ 3}

{ x ∈ ℝ | 0 ≤ x ≤ 1}

Let set A = { a, b, c } and set Z = { x, y, z }. Using these sets, answer the following question: Identify one subset of set A that has cardinality 2.

{a,b} {b, a} {b, c} {c, a} {c, b} {a,c}

Conjunction Elimination, Conjunction Introduction, Constructive Dilemma, Disjunction Introduction, Disjunctive Syllogism, Hypothetical Syllogism, Modus Tollens, Modus Ponens Complete the logical argument below. In the first 4 steps, indicate a propositional expression that accurately represents each given premise. Use the underlined letters (c, d, e and s) to represent the individual propositions needed. In the remaining steps, indicate an assertion supported by the stated reason. The Rules of Inference list is given for reference. Assertion | Reason c ∨ e | Premise 1: Tom is a computer science major or an engineering major c → s | Premise 2: Tom being smart is necessary for Tom to be a computer science major ¬ d → ¬ s | Premise 3: If Tom does not know discrete math Tom is not smart ____________________________________ | Premise 4: Tom is not an engineering major ¬ d ¬ c ¬ s ¬ e

¬ e

With the universe of discourse for x as the set of all current FSU students and the universe of discourse for y as the set of all 50 states in the United States, we define the following predicate: T(x, y) represents "Person x wants to travel to state y." Indicate which symbolic expression accurately uses quantifiers with the given predicate to express this statement: "There is some current FSU student who wants to travel to all 50 states in the US." ∀ y ∃ x T( x, y ) ∃ x ∀ y T( x, y ) ∃ y ∀ x T( x, y ) ∀ x ∃ y T( x, y )

∃ x ∀ y T( x, y )

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all states in the United States, we define the following predicates: F(x) is "x is a current FSU student," G(x) is "x is a graduate of FSU," R(x, y) is "x is a resident of y." Using the given definitions of these predicates, match each English sentence with the symbolic expression that accurately represents the sentence. There is a person who is a current FSU student who is not a resident of Florida. ∃x [R( x, Florida ) ∧ ¬ F( x )] ∃x [¬ F( x ) ∧ ¬R( x, Florida ) ] ∃x [ R( x, Florida ) ∧ G( x ) ] ∃x [ R( x, Florida ) ∧ F( x ) ] ∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ] ∃x [¬ F( x ) ∧ G( x ) ∧ ¬ R( x, Florida ) ] ∃x [ F( x ) ∧ ¬ R( x, Florida ) ]

∃x [ F( x ) ∧ ¬ R( x, Florida ) ]

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all states in the United States, we define the following predicates: F(x) is "x is a current FSU student," G(x) is "x is a graduate of FSU,"R(x, y) is "x is a resident of y." Using the given definitions of these predicates, match each English sentence with the symbolic expression that accurately represents the sentence. There is a Florida resident who is a current FSU student. ∃x [R( x, Florida ) ∧ ¬ F( x )] ∃x [¬ F( x ) ∧ ¬R( x, Florida ) ] ∃x [ R( x, Florida ) ∧ G( x ) ] ∃x [ R( x, Florida ) ∧ F( x ) ] ∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ] ∃x [¬ F( x ) ∧ G( x ) ∧ ¬ R( x, Florida ) ]

∃x [ R( x, Florida ) ∧ F( x ) ]

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all states in the United States, we define the following predicates: F(x) is "x is a current FSU student," G(x) is "x is a graduate of FSU,"R(x, y) is "x is a resident of y." Using the given definitions of these predicates, match each English sentence with the symbolic expression that accurately represents the sentence. Some residents of Florida are graduates of FSU. ∃x [R( x, Florida ) ∧ ¬ F( x )] ∃x [¬ F( x ) ∧ ¬R( x, Florida ) ] ∃x [ R( x, Florida ) ∧ G( x ) ] ∃x [ R( x, Florida ) ∧ F( x ) ] ∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ] ∃x [¬ F( x ) ∧ G( x ) ∧ ¬ R( x, Florida ) ]

∃x [ R( x, Florida ) ∧ G( x ) ]

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all states in the United States, we define the following predicates: F(x) is "x is a current FSU student," G(x) is "x is a graduate of FSU,"R(x, y) is "x is a resident of y." Using the given definitions of these predicates, match each English sentence with the symbolic expression that accurately represents the sentence. There is someone who is a Florida resident who is not a current FSU student and not an FSU graduate. ∃x [R( x, Florida ) ∧ ¬ F( x )] ∃x [¬ F( x ) ∧ ¬R( x, Florida ) ] ∃x [ R( x, Florida ) ∧ G( x ) ] ∃x [ R( x, Florida ) ∧ F( x ) ] ∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ] ∃x [¬ F( x ) ∧ G( x ) ∧ ¬ R( x, Florida ) ]

∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ]

To disprove a statement of the form ∀x [P(x) ⟶ Q(x)] by counter-example, what do you need to demonstrate? ∃x ¬ [P(x) ∧ Q(x)] ∃x [¬P(x) ∧ Q(x)] ∃x [P(x) ∧ ¬ Q(x)] ∃x [P(x) ∧ Q(x)]

∃x [P(x) ∧ ¬ Q(x)]

With the universe of discourse for x as the set of all people alive in the world and the universe of discourse for y as the set of all states in the United States, we define the following predicates: F(x) is "x is a current FSU student," G(x) is "x is a graduate of FSU,"R(x, y) is "x is a resident of y." Using the given definitions of these predicates, match each English sentence with the symbolic expression that accurately represents the sentence. Some Florida residents are not current FSU students. ∃x [R( x, Florida ) ∧ ¬ F( x )] ∃x [¬ F( x ) ∧ ¬R( x, Florida ) ] ∃x [ R( x, Florida ) ∧ G( x ) ] ∃x [ R( x, Florida ) ∧ F( x ) ] ∃x [ R( x, Florida ) ∧ ¬ F( x ) ∧ ¬G( x ) ] ∃x [¬ F( x ) ∧ G( x ) ∧ ¬ R( x, Florida ) ]

∃x [R( x, Florida ) ∧ ¬ F( x )]