Midterm

p ⟶ q

(TFTT)

¬q⟶¬p

(TFTT)

Choose the best answer for f(n) = ⌈n/2⌉, where n ∈ Z, note that ⌈n/2⌉ is the smallest integer that is ≥ n/2

This is an onto function

Choose the best answer for f(x) = (x+1)/(x+2), where x ∈R

This is not a function

Choose the best answer for f(x) = (x^2+1)/(x+2), where x ∈R

This is not a function

How would you prove that ∀n∈N, 2n+1 is a prime number?

This is not true.

Create a truth table to determine if [¬q∧(p⟶q)]⟶¬p is a tautology. Mark true if it is and false if it is not. (Note: Don't guess! Write out the truth table on paper before you answer.)

True

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∀x∀y∃z(z=(x+y)/2)

True

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∀x∃y(x+y=1)

True

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∀x∃y(x^2=y)

True

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∃x(x^2=2)

True

Determine the truth value of ∀n(n^2≥0) if the universe of discourse for all variables is the set of all integers.

True

Determine the truth value of ∀n(n^2≥n) if the universe of discourse for all variables is the set of all integers.

True

Determine the truth value of ∀n∃m(n+m=0) if the universe of discourse for all variables is the set of all integers.

True

Determine the truth value of ∃n∀m(nm=m) if the universe of discourse for all variables is the set of all integers.

True

Determine the truth value of ∃n∃m(n^2+m^2=5) if the universe of discourse for all variables is the set of all integers.

True

Determine whether this statement is true or false: {ϕ}∈{{ϕ}}

True

Determine whether this statement is true or false: {ϕ}⊂{ϕ,{ϕ}}

True

Determine whether this statement is true or false: ϕ∈{ϕ}

True

Is Symmetric Difference associative? That is, if A, B and C are sets, is it true that A⊕(B⊕C)=(A⊕B)⊕C ?

True

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: P(0,8)

True

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: ∃n¬P(2,n)

True

Universal set, set containing all elements under consideration

U

The proposition: "P(x) is true for all values of x in U."

Universal Quantification of P(x)

The domain of a variable in a propositional function.

Universe of Discourse

Set of all real numbers.

R

Fill in the blanks to complete the proof that (A-B)-C ⊆ A-C for arbitrary sets A,B and C. Proof: We must show that for any e∈(A-B)-C, then e∈A-C. Let e be an arbitrary element of ______1____. Then e ∈___2_______ and e∉_____3_____ . If e ∈____4______, then e ∈______5____ and e∉_____6_____. Thus e ∈_____7_____ and e∉____8______, which means that e must be an element of ____9______. Therefore (A-B)-C ⊆ A-C.

1. (A-B)-C 2. A-B 3. C 4. A-B 5. A 6. B 7. A 8. C 9. A-C

Let A = {a,b,c,d,e}, B = {a,b,c,d,e,f,g,h} and C = {f,g,h}. Fill in each blank with an identical set chosen from: A, B, C, Empty Set, or No Match: 1. A∪B is the same as 2. A∩B is the same as 3. A−B is the same as 4. B−A is the same as 5. A∪C is the same as 6. A∩C is the same as 7. A⊕B is the same as 8. B⊕A is the same as

1. B 2. A 3. Empty Set 4. C 5. B 6. Empty Set 7. C 8. C

Justify each step with the name of the logical equivalence that was used to prove that (¬q∧(p⟶q))⟶¬p is a tautology. 1. (¬q∧(p⟶q))⟶¬p 2. (¬q∧(¬p∨q))⟶¬p 3. ¬(¬q∧(¬p∨q))∨¬p 4. (¬¬q∨¬(¬p∨q))∨¬p 5. (q∨¬(¬p∨q))∨¬p 6. (q∨(¬¬p∧¬q))∨¬p 7. (q∨(p∧¬q))∨¬p 8. ((q∨p)∧(q∨¬q))∨¬p 9. ((q∨p)∧(T))∨¬p 10. (q∨p)∨¬p 11. q∨(p∨¬p) 12. q∨T 13. T

1. Initial Statement Given 2. Implication Equivalence 3. Implication Equivalence 4. DeMorgan 5. Double Negative 6. DeMorgan 7. Double Negative 8. Distributive 9. Negation 10. Identity 11. Associative 12. Negation 13. Domination

Determine whether f: Z x Z -->Z is onto if 1. f(m,n) = 2m-n 2. f(m,n) = m^2 - n^2 3. f(m,n) = m+n+1 4. f(m,n) = |m| - |n| 5. f(m,n) = m^2-4

1. Onto 2. Not onto 3. Onto 4. Onto 5. Not onto

Determine whether each of the following functions from the set {a,b,c,d} to itself is one-to-one. 1. f(a)=b, f(b)=a, f(c)=c, f(d)=d 2. f(a)=b, f(b)=b, f(c)=d, f(d)=c 3. f(a)=d, f(b)=b, f(c)=c, f(d)=d

1. one-to-one 2. not one-to-one 3. not one-to-one

Let A = {a, b, c, d, e}, B = {d, e, f, g} and the Universal Set, U, be the letters a through h in the English alphabet. Match each set definition with the set they describe. 1) {x : x∈U ∧ x∉A} 2) {x : x∈A ∨ x∈B} 3) {x : x∈A ∧ x∈B} 4) {x : x∈A ∧ x∉B} 5) {x : x∈A ⊕ x∈B}

1. {f, g, h} 2. {a, b, c, d, e, f, g, h} 3. {d, e} 4. {a, b, c} 5. {a, b, c, f, g}

If f(x) = x^2+1 and g(x) = x+2 then (g∘f)(3)=

12

Choose the best answer for f(x) = -3x + 4, where x ∈ R

This is a one-to-one and onto function.

The cardinality of the power set of A = {r,s,t,u,w} is equal to what value?

32

Choose the best answer for f(x) = 2x + 1, where x ∈R

This is a one-to-one and onto function.

Match each statement 1-6 with an equivalent statement 1. ¬∃x∃yP(x,y) 2. ¬∀x∃yP(x,y) 3. ¬∀x∀yP(x,y) 4. ¬∃x¬∃yP(x,y) 5. ¬∃y(Q(x,y)∧∀x¬R(x,y)) 6. ¬∀y∀x(Q(x,y)∨R(x,y))

6. ∃y∃x(¬Q(x,y)∧¬R(x,y)) 2. ∃x∀y¬P(x,y) 5. ∀y(¬Q(x,y)∨∃xR(x,y)) 3. ∃x∃y¬P(x,y) 1. ∀x∀y¬P(x,y) 4. NO Match

{x : x ∈ A ∧ x ∉ B}

A - B

(A⊆B)∧(B⊆A)

A = B

On a remote island there live Knights and Knaves. Knights always tell the truth and Knaves always lie.You meet two people on the island—A and B.A says: "B and I are both Knights or both Knaves."B says: "I and A are the same"From their statements, determine who is a knight and who is a knave.

A and B are both Knights

On a remote island there live Knights and Knaves. Knights always tell the truth and Knaves always lie. You meet two people on the island, A and B. A says: "We are both Knaves." B says nothing. Using the truth table technique we went over in class determine, if possible, which group A and B belong to.

A is a Knave and B is a Knight.

Mark the appropriate minterms that would be included in the disjunctive normal form of(p⟶q)⟶r

A. (p∧q∧r) C. (p∧¬q∧r) D. (p∧¬q∧¬r) E. (¬p∧q∧r) G. (¬p∧¬q∧r)

Mark the appropriate minterms that would be included in the disjunctive normal form of p⟶(q∧r).

A. (p∧q∧r) E. (¬p∧q∧r) F. (¬p∧q∧¬r) G. (¬p∧¬q∧r) H. (¬p∧¬q∧¬r)

Given the proposition p⟶(q∧r), match each variation of the proposition to its related implication.

A. ¬(q∧r)⟶¬p (equivalent to original proposition) B. (q∧r)⟶p (equivalent to the inverse of the original proposition) C. ¬p⟶¬(q∧r) (equivalent to the contrapositive of the original proposition) D. ¬p∨(q∧r) (equivalent to the converse of the original proposition)

Choose the best answer for f(x) = 2x +1, where x ∈R

This is a one-to-one and onto function.

Choose the best answer for f(x) = x^3, where x ∈R

This is a one-to-one and onto function.

{x : x ∈ A ∨ x ∈ B}

A∪B

(A⊆B)∧(A≠B)j)

A⊂B

∀x(x∈A→x∈B)i)

A⊆B

{x : x ∈ A ⊕ x ∈ B}

A⊕B

Number of elements in a set, S, denoted by |S|.

Cardinality of a Set

{x : x ∈ U ∧ x ∉ A}

Complement of A

Compound proposition that is always false regardless of the truth values of the propositions in it.

Contradiction

approach for a proof: The product of two irrational numbers is irrational.

Counter Example

approach for a proof: The sum of two even integers is even.

Direct Proof

approach for a proof: The sum of two odd integers is even.

Direct Proof

The proposition: "There exists an element, x, in U such that P(x) is true."

Existential Quantification of P(x)

Put the statements shown below in order to complete a proof by contradiction that the product of a nonzero rational number and an irrational number is irrational. I have provided the first two statements. (Hint: Write out the proof on paper for yourself before you try to order the statements.) Proof: Assume that the product of a nonzero rational number and an irrational number is rational. Let x be a nonzero rational number such that r = a/b where a and b are nonzero integers, and q be an irrational number. Statements to complete the proof: (A) But cb and da are both integers since they are the product of integers, so cb/da is a rational number. (B) It follows that r*q = (a/b)q = c/d. (C) However this means that we have a irrational number, q, equal to a rational number cb/da. This is a contradiction. (D)Solving for q, we get q = cb/da. (E) Thus our assumption that r*q was rational was false, so r*q must be irrational. (F) By our assumption, r*q is rational so r*q = c/d where c and d are integers and d is nonzero.

F. B. D. A. C. E.

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∀x∃y((x+y=2)∧(2x−y=1))

False

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∃x(x^2=−1)

False

Determine the truth value of the following statement if the universe of discourse is the set of real numbers. ∃x∃y((x+2y=2)∧(2x+4y=5))

False

Determine the truth value of ∃n(n^2=2) if the universe of discourse for all variables is the set of all integers.

False

Determine whether the function f:Z x Z --> Z is onto if f (m,n) = m^2 + n^2

False

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: P(1,8)

False

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: ∀m∀nP(m,n)

False

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: ∀m∃nP(m,n)

False

Let P(m,n) be the statement m+n≤mn where the universe of discourse for both variables is the set of non-negative integers. State whether the following proposition is true or false: ∃m∀nP(m,n)

False

Choose the best answer for f(x) = x^5 + 1, where x ∈R

This is a one-to-one and onto function.

A set of logical operators is called ________ if every compound proposition is logically equivalent to a compound proposition involving only these logical operators.

Functionally Complete

approach for a proof: (n is an integer and n**3 + 5 is odd) --> n is even

Indirect Proof

approach for a proof: If a**2 is even, then a is even.

Indirect Proof

approach for a proof: If n is an integer and 3n+2 is odd, then n is odd.

Indirect Proof

Choose the best answer for f(n) = n^3, where n ∈ Z

This is a one-to-one function

Fill in the blanks below to complete the proof that the product of two rational numbers is rational. Choose your answers from the following options. Enter them exactly like I have them written! a/b a*b c/d c*d (a/b)(c/d) (ab)(cd) (ac)/(bd) (ac)(bd) integer integers rational rationals s*t s/t

Let s,t be RATIONAL numbers. Then there exists INTEGERS a,b,c and d where b,d are not equal to 0 such that s= A/B and t = c/d. It follows that S*T = (A/B)(C/D) = (AC)/(BD). Then S*T is RATIONAL because ac and bd are both INTEGERS and bd is not equal to zero. Thus the product of two rational numbers is rational. Q.E.D.

Set of Natural Numbers

N

Set of Positive Integers

P or Z+

Set of all subsets of A

Power Set

approach for a proof: The square root of 3 is irrational.

Proof by Contradiction

A statement that is either true or false, but not both.

Proposition

Propositions that contain variables.

Propositional Functions or Predicates

Collection of objects, usually denoted by a capital letter.

Set

Compound proposition that is always true regardless of the truth values of the propositions in it.

Tautology

Choose the best answer for f(n) = n^2 + 1, where n ∈ Z

This is a function

Choose the best answer for f(x) = (x^2+1)/(x^2+2), where x ∈R

This is a function

Choose the best answer for f(x) = -3x^2 +7, where x ∈R

This is a function

Choose the best answer for f(x) = x^2 + 1, where x ∈R

This is a function

Choose the best answer for f(x) = x^2 +1, where x ∈R

This is a function

Choose the best answer for f(n) = n - 1, where n ∈ Z

This is a one-to-one and onto function.

Set of all positive, negative and zero integers.

Z

The proposition that is true when p and q have the same truth values and is false otherwise.

p <--> q

p ⟷ q

p if and only if q (TFFT)

(p ∧ ¬q) V (¬p ∧ q) is equivalent to which of the following propositions?

p⊕q

Converse of p --> q

q -> p

Fill in the blanks to describe the set T = {0,1,4,9,16} using set builder notation. t = m2: ________ ∈ ________ ∧ m < ________

t = m2: m ∈ N ∧ m < 5

p ⊕ q (exclusive or of p and q)

the proposition "p XOR q," which is true when exactly one of p and q is true (FTTF)

p ∧ q (conjunction of p and q)

the proposition "p and q," which is true if and only if both p and q are true (TFFF)

Empty Set

{} or ϕ

Inverse of p --> q

¬p -> ¬q

Below are the steps for proving that ¬p⟶(q⟶r)⟺q⟶(p∨r). Justify each step with the name of the logical equivalence that was used.

¬p⟶(q⟶r) 1.) ⟺¬(¬p)∨(q⟶r) (Implication Equivalence) 2.) ⟺¬(¬p)∨(¬q∨r) (Implication Equivalence) 3.) ⟺p∨(¬q∨r) (Double Negation) 4.) ⟺(p∨¬q)∨r (Associative) 5.) ⟺(¬q∨p)∨r (Commutative) 6.) ⟺¬q∨(p∨r) (Associative) 7.) ⟺q⟶(p∨r) (Implication Equivalence)

Contrapositive of p --> q

¬q -> ¬p