MAT 243 final

Generally, the sum of two 3-digit hexadecimal numbers is a hexadecimal number with

3 or 4 digits.

Generally, the product of a 2-digit and a 3-digit hexadecimal number is a hexadecimal number with

4 or 5 digits.

The binary form of a 3-digit hex number has

9, 10, 11 or 12 binary digits.

Check all true statements, and only those. Conjunction can be expressed using disjunction and negation. Conjunction can be expressed using conditional and negation. Disjunction can be expressed using conjunction and negation. Disjunction can be expressed using conditional and negation. Conditional can be expressed using disjunction and negation. Conditional can be expressed using conjunction and negation.

Conjunction can be expressed using disjunction and negation. Conjunction can be expressed using conditional and negation. Disjunction can be expressed using conjunction and negation. Disjunction can be expressed using conditional and negation. Conditional can be expressed using disjunction and negation. Conditional can be expressed using conjunction and negation.

True or false? If P(n) is a summation formula involving a sigma sum, we always prove the statement P(n+1) by taking the statement P(n) and adding n+1 to both sides.

False

True or false? If a is a positive real number, and x and y are real numbers, then aˣ⁺ʸ = aˣ + aʸ.

False

True or false? If x is a real number, then 2·3ˣ = 6ˣ.

False

True or false? In the inductive step, we justify the assumption P(n) by referring to the base case. (Example: "Since P(0) is true, we can now assume that P(n) is true for some n...")

False

True or false? We always prove the statement P(n+1) by taking the statement P(n) and adding n+1 to both sides.

False

True or false? a² + b² = c² is a proposition.

False It's not a proposition because a, b and c are undefined (free) variables.

True or false? The equation p = 5 + 7 = 12 defines p to be the proposition that 5 + 7 equals 12.

False The equation defines p to be the quantity 12. If you wish to define p as the proposition that 5 + 7 = 12, you have to use appropriate parentheses: p = ( 5 + 7 = 12 ).

Determine the converse of If f is differentiable, then f is continuous.

If f is continuous, then f is differentiable.

Check the expressions, and only those, that are equivalent to the conditional statement p → q. If p, then q p only if q q only if p p is necessary for q q when p q is a necessary condition for p q if p p is a sufficient condition for q q is a sufficient condition for p ¬p ∨ q ¬(p ∧ ¬q) ¬(p ∨ ¬q) q→p p ∧ ¬q

If p, then q p only if q q when p q is a necessary condition for p q if p p is a sufficient condition for q ¬(p ∧ ¬q) ¬(p ∨ ¬q)

Mark all true statements. If p,q and r are distinct primes, then lcm(pq, qr) = prq. The gcd of two distinct primes is 1. If p,q and r are distinct primes, then gcd(pq, qr) = q. Given the prime factorizations of positive integers a and b, we can easily find the gcd and the lcm of a and b. The lcm of two distinct primes is their product.

If p,q and r are distinct primes, then lcm(pq, qr) = prq. The gcd of two distinct primes is 1. If p,q and r are distinct primes, then gcd(pq, qr) = q. Given the prime factorizations of positive integers a and b, we can easily find the gcd and the lcm of a and b. The lcm of two distinct primes is their product.

Find the inverse of "If x > 0 and y < 0, then xy < 0. "

If x ≤ 0 or y ≥ 0, then xy ≥ 0.

Consider the following argument: All wizards can do magic. Muggles can't do magic. Peter is a muggle. Therefore, Peter is not a wizard. Several students in Hogwarts formalized this argument (without using magic). Here is Luna's attempt: Luna: Let's define the predicates: W= " is a wizard", M= " is a muggle", A=" can do magic". Universe of discourse: all people (magical and non-magical) 1. ∀xW(x)→ A(x) (assumption) 2. ∀xM(x) → ¬A(x) (assumption) 3. M(Peter) (assumption) 4. M(Peter)→ ¬

Luna's argument is invalid. Luna's argument is invalid, because in 1 the quantifier only refers to W(x) and not to A(x) A(x) is not bound to any quantifier. Thus, 1 is not a proposition. Luna's argument is invalid, because in 2 the quantifier only refers to M(x) and not to ¬A(x) ¬A(x) is not bound to any quantifier. Thus, 2 is not a proposition.

Check all logical expressions that are logically equivalent to the negation of p → q, and only those. ¬p → q p → ¬q ¬p → ¬q ¬q → p q → ¬p ¬q → ¬p None of those.

None of those.

We always start the inductive step with the assumption that

P(n) is true for some n.

An inductive proof that P(n) is true for all n always starts with the base case. What is the base case?

The base case is always P(n) for the lowest n value for which P(n) is defined.

What is the correct meaning of the expression p→q∨r→s ?

The expression has no unique meaning. Disjunction enjoys a higher operator precedence than the conditional. Therefore, p→q∨r→s means p→(q∨r)→s. There is no universal convention however whether the conditional is left- or right-associative. Thus, p→(q∨r)→s could mean (p→(q∨r))→s or p→((q∨r)→s).

Your job is to write a basic blurring algorithm for a video driver. The algorithm will do the following: it will go through all pixels on the screen and for each pixel, compute the average intensity value (in red, green and blue separately) of the pixel and its 8 neighbors. (At the edges of the screen, there are fewer neighbors for each pixel.) Let's say the number of pixels on the screen is n. Then what is the order of the number of arithmetic operations (additions and divisions) required?

The number is order of n .

What is the negation of "All MAT 243 students know logic "?

There is at least one MAT 243 student who doesn't know logic.

"a is necessary for b" is logically equivalent to "a only if b".

false

To express the fact that the two inequalities 2 < x < 4 6 < 3x < 12 are logically equivalent, we may write 2 < x < 4 = 6 < 3x < 12.

false You can say 2 < x < 4 if and only if 6 < 3x < 12 or 2 < x < 4 is equivalent to 6 < 3x < 12.

In the inductive step, we show that

if P(n) is true, then P(n+1) is true.

p ∧ p ≡ p is..

one of the idempotent laws.

Casey applies for a scholarship and is interviewed. He tells the selection committee: I had very good grades in high school. Also, the teachers liked me. The selection committee writes down: had good grades in high school. What rule of inference did the committee use?

simplification

"a is necessary for b" is logically equivalent to "b is sufficient for a".

true

"a is sufficient for b" is logically equivalent to "a only if b".

true

For all variables P with two variables, ∃x∀y P(x,y) → ∀y∃x P(x,y).

true

For any predicates P and Q of one variable, ∀xP(x) ∧ ∀xQ(x) → ∀x (P(x) ∧Q(x)).

true

For any predicates P and Q of one variable, ∃x (P(x) ∧Q(x)) → ∃xP(x) ∧ ∃xQ(x).

true

True or False? In hexadecimal, you divide a positive integer by 0x10 by separating the last digit r from the rest of the number q. r is the remainder, q the quotient of the division.

true

The domain restricted universal statement ∀x>0 (x² > 2) means the same as..

∀x ( x>0 → x² > 2 )

What is the negation of ∃x>1 ( x³ > 2) ?

∀x > 1 ( x³ ≤ 2)

Check each true statement, and only the true statements. The domain for x is the set {0}, the domain for y is the set {-1,1}. ∀x ∃y (x²+y²=1) ∃x ∀y (x²+y²=1) ∀x ∀y (x²+y²=1) ∃x ∃y (x²+y²=1)

∀x ∃y (x²+y²=1) ∃x ∀y (x²+y²=1) ∀x ∀y (x²+y²=1) ∃x ∃y (x²+y²=1)

Let B and H be predicates defined on the set of all animals by B = "is a bunny" and H = "hops". Use quantifiers and the given predicates to express the following statement. Every bunny hops but not every animal that hops is a bunny.

∀x(B(x) → H(x)) ∧ ∃x (H(x) ∧ ¬B(x))

Check all true statements and only those. P is any predicates with two variables. ∀x∀yP(x,y) is equivalent to ∀y∀xP(x,y) ∃x∃yP(x,y) is equivalent to ∃y∃xP(x,y) ∀x∀yP(x,y) →∃x∃yP(x,y)) ∃y∃xP(x,y)→∀y∀xP(x,y) ∀x∀yP(x,y)→∀x∃yP(x,y) ∀x∀yP(x,y)→∃y∀xP(x,y)

∀x∀yP(x,y) is equivalent to ∀y∀xP(x,y) ∃x∃yP(x,y) is equivalent to ∃y∃xP(x,y) ∀x∀yP(x,y) →∃x∃yP(x,y)) ∀x∀yP(x,y)→∀x∃yP(x,y) ∀x∀yP(x,y)→∃y∀xP(x,y)

Check each true statement, and only the true statements. The domain for all variables is the set of integers.

∀y ∃x (x+2y=1)

Check all true statements and only those. The domain is the set of real numbers. ∃x ( x > 1 → x² < 0 ) ∀x( x > 1 → x² < 0 ) ∀x( x > 1 → x² > 1 )

∃x ( x > 1 → x² < 0 ) ∀x( x > 1 → x² > 1 ) ∃x ( x > 1 → x² <0 ) is true because x>1 → x²<0 is true for x=0. This real number makes the premise of the conditional false, and thus the conditional true.

The domain restricted existential statement ∃x>0 (x² > 2) means the same as.. ∃x ( x>0 ∧ x² > 2 ) ∃x ( x>0 → x² > 2 )

∃x ( x>0 ∧ x² > 2 )

With the domain of discourse the positive integers, express "there is a smallest positive integer" as a formal statement.

∃x ∀y ( x ≤ y )

Negate the statement ∀y∃x( (x>y) → (x+y > 0) ).

∃y∀x( (x>y) ∧ (x+y ≤ 0) ).

Let p = " I walk to school every day", q = "I use the swimming pool" and r = "I build cardiovascular health." Express the following statement as a combination of p,q and r and logic symbols. "To build cardiovascular health, it is sufficient for me to walk to school every day or to use the swimming pool."

( p∨q ) → r

When we apply the distributive law to (a∧b) ∨c, we get

(a∨c) ∧ (b∨ c)

When we apply the definition of conditional and De Morgan to (p → q) → q, we get

(p ∧ ¬q) ∨q

In hexadecimal, you multiply a positive integer by 0x10 by appending the digit

0

A non-negative integer x can be represented in hexadecimal with up to 4 digits if and only if..

0 ≤ x < 16⁴.

Suppose you apply the Euclidean algorithm to two positive integers a, b. You only know that a is a number with 1000 decimal digits.The value of b on the other hand is given: b = 3. Then we can be certain that the Euclidean algorithm will end with a zero remainder in how many steps? Enter the lowest number we can be certain of.

3

Refering again to the blur algorithm described previously, by what factor does the number of operations increase when instead of applying it at full HD resolution (1920 x 1080), we apply it at UHD resolution (3840 x 2160) ?

4

What do you get when you divide a 64-bit number by 2? If there are several correct answers, select the one with the lowest number of quotient bits.

A 63-bit quotient and a remainder bit.

Check all true statements. A conditional is equivalent to its converse. A conditional is equivalent to its contrapositive. A conditional is equivalent to its inverse. Converse and inverse of a conditional are equivalent. Converse and contrapositive of a conditional are equivalent. Inverse and contrapositive of a conditional are equivalent.

A conditional is equivalent to its contrapositive. Converse and inverse of a conditional are equivalent.

You may have noticed that in practice problems related to order, logarithms are usually just "log". As you know from algebra, there is more than one logarithm. For each positive number b, there is a base-b logarithm. They are all different- the base-2 logarithm of 8 is 3, while the base 10 logarithm of 8 is 0.90308998699... . There is also a base-e logarithm, called the natural logarithm and usually written as "ln". This begs the question of how it could be justified to just say that a function

Actually, "f is order of log(n)" means the same thing regardless of the base. More precisely, if a and b are two positive real numbers, then f is of order base-a log of n if and only if f is of order base-b log of n. The reason for that lies in the change of base formula, which says that any two log functions are in a constant positive factor relationship with each other. Order relations between two function do not change when you change a constant positive factor in one of the functions.

Identify the following as a valid argument or a fallacy. If the argument is valid, identify the argument form. "2 is an even number. 2 does not have proper divisors. Therefore 2 is an even prime number."

Conjunction

Find the inverse, converse, contrapositive and the negation of the following conditional statement: "If x ≥ 0 and y ≥ 0, then x+y ≥0".

Contrapositive: If x+y < 0, then x<0 or y<0 converse: If x+y>=0, then x>=0 and y>=0 Inverse:If x<0 or y<0, then x+y<0 Negation: x>=0 and y >=0 and x+y>=0

Check all true statements.' You can convert a number from decimal to binary by replacing each decimal digit separately by its corresponding binary representation. Expressed in base-n, the integer n is "10". In octal (base-8), every digit is 0,1,2,3,4,5,6 or 7. If k is an integer greater than 1 and n is a positive integer that is not a power of k, then n has ⌈logₖ(n)⌉ digits in base k. In ternary (base-3), every digit is a 0,1 or 2. In base-b, it is easy to see whether an integer is a

Expressed in base-n, the integer n is "10". In octal (base-8), every digit is 0,1,2,3,4,5,6 or 7. If k is an integer greater than 1 and n is a positive integer that is not a power of k, then n has ⌈logₖ(n)⌉ digits in base k. In ternary (base-3), every digit is a 0,1 or 2. In base-b, it is easy to see whether an integer is a multiple of b. Its last digit is zero in that case. The fast modular exponentiation algorithm takes advantage of the binary representation of the exponent. You can convert a number from binary to octal by grouping the digits ("bits") of the binary number into groups of 3, going from right to left. If the number of bits is not a multiple of 3, you may have to add one or two leading 0 bits on the left side. Then you convert each group of 3 bits into one octal digit. In duodecimal (base-12), every digit is 0,1,2,3,4,5,6,7,8,9,A or B. Among all base-b representations of a positive integer n, the binary one is always at least as long as any other (in terms

Identify the following as a valid argument or a fallacy. If the argument is valid, identify the argument form. If you can vote, then you 18 or older. Kylie cannot vote. Therefore, Kylie is younger than 18.

Fallacy

Identify the following as a valid argument or a fallacy. If the argument is valid, identify the argument form. Every bug is an insect. Wizzy-Fizz is an insect. Therefore, Wizzy-Fizz is a bug.

Fallacy of affirming the conclusion

Judge the following reasoning as true or false.You are working for a tech company. You came up with an algorithm that performs a needed operation on n inputs in order of n operations. Someone else in the company simultaneously came up with an algorithm that performs the same task in order of log(n) operations. Then it follows necessarily that your algorithm will require more operations to perform this task and is therefore inferior and should not be used.

False

True or false? The conclusion of the inductive step, P(n+1), is shown by substituting n+1 for the n in the statement P(n).

False

True or false? The truth value of If the sun shines today, then it is warm. is undefined if the sun is not shining today.

False If the premise of a conditional is false, then the conditional is true.

Let H be the predicate defined on the set of all people by H(x) = "x is happy". Which of the following statements are acceptable translations of the formal statement ∃x H(x) into standard English? Happy people exist. There exists a person x such that x is happy. There exists a happy person. There are happy people. There is at least one happy person.

Happy people exist. There exists a happy person. There are happy people. There is at least one happy person. The existential quantifier can be translated as "there is" or "there are". It does not distinguish between existence and unique existence. In standard English, we don't use variable names to refer to objects.

Mark all true statements. If a,b,q and r are integers and a= bq + r, then gcd(a,b) = gcd(b,r). The last nonzero remainder in the Euclidean algorithm is the gcd. The first remainder in the Euclidean algorithm is an upper limit for the number of steps until the algorithm terminates. Remainders in the Euclidean algorithm decrease strictly. It is possible for the Euclidean algorithm to terminate in one step.

If a,b,q and r are integers and a= bq + r, then gcd(a,b) = gcd(b,r). The last nonzero remainder in the Euclidean algorithm is the gcd. The first remainder in the Euclidean algorithm is an upper limit for the number of steps until the algorithm terminates. Remainders in the Euclidean algorithm decrease strictly. It is possible for the Euclidean algorithm to terminate in one step.

Check all statements that are true. If an integer divides two numbers, it also divides their sum. When you perform division by 3 with remainder, the remainder is one of the integers 0,1,2. When you perform division by 5 with remainder, the remainder is an integer from -5 to 5. Adding two integers and then taking the remainder produces the same result as taking their remainders first, then adding them, and then applying the remainder operation once more. If an integer divides two numbers, it

If an integer divides two numbers, it also divides their sum. When you perform division by 3 with remainder, the remainder is one of the integers 0,1,2. Adding two integers and then taking the remainder produces the same result as taking their remainders first, then adding them, and then applying the remainder operation once more. If an integer divides two numbers, it also divides their difference. Saying that a divides b is the same as saying that b is a multiple of a. If a and b are positive integers, and a = bq + r is the decomposition of a given by the division algorithm, then q can be found as the floor of a/b, and then r can be found as r = a - bq. If an integer a divides an integer b, then a also divides any multiple of b.

Identify the following as a valid argument or a fallacy. If the argument is valid, identify the argument form. Every computer science student takes a discrete mathematics course. Sydney is a computer science student. Therefore Sydney takes a discrete mathematics course.

Modus ponens

Identify the following as a valid argument or a fallacy. If the argument is valid, identify the argument form. Today I go for a walk and call my sister. Therefore, I call my sister today.

Simplification

We want to prove by induction that n² + n is even for all positive integers n. The base case is: 1² +1 = 2 is even. Select the proper inductive hypothesis. It is left up to you to determine whether there are multiple correct answers.

Suppose we have proven that n² + n is even for some arbitrary positive integer n.

Mark all true statements The number of prime numbers between 1 and 100 is greater than the number of prime numbers between 1000 and 1100. There is a largest prime, and it is about the size of Graham's number. If the positive integers a and b are relatively prime, then so are a+1 and b+1. The sum of two primes is prime. The number n can have at most the floor of base-2 log of n many prime factors, if prime factors are counted with repetition (i.e. 2*3*3*5 has 4 prime factors.) If p is prime

The number of prime numbers between 1 and 100 is greater than the number of prime numbers between 1000 and 1100. The number n can have at most the floor of base-2 log of n many prime factors, if prime factors are counted with repetition (i.e. 2*3*3*5 has 4 prime factors.) If there are only finitely many primes, then 1=2. If p and q are distinct primes, then they are also relatively prime to each other. To test whether 101 is prime, you only need to divide it by 2,3,5 and 7. If it is not divisible by any of those numbers, then it is prime. The sequence of prime numbers follows no exact pattern. If p is prime, then p+2 may or may not be prime, i.e. there exist primes p such that p+2 is prime, and there exist primes p such that p+2 is not prime.

Proof: let P ( n ) = ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 .Base case: P(1) = 1/2. Inductive step: suppose P(n) has already been proven for some arbitrary n. The statement P(n+1) is P ( n + 1 ) = ∑ k = 1 n + 1 1 k ( k + 1 ) = 1 − 1 n + 2This concludes the proof by induction. P ( n + 1 ) = ∑ k = 2 n + 1 1 ( k + 1 ) ( k + 2 ) = 1 − 1 n + 2 . There is nothing wrong with this proof. The proof writer confused stating P(n+1) with showing that it must be true, given P(n) is true. The pr

The proof writer confused stating P(n+1) with showing that it must be true, given P(n) is true. The proof abuses the notation P(n) to refer to both the common value of the two sides of the equation to be proved and the statement that the two sides are indeed equal as the notation was introduced in the lecture. Furthermore, it does not make sense to define P(n) as the common value of the two sides, because it assumes the conclusion, that the two sides are equal. At the very least, the definition of P(n) in the first line should have used parentheses: P ( n ) = ( ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 ) .P ( n ) = ( ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 ) .P ( n ) = ( ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 ) .P ( n ) = ( ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 ) .P ( n ) = ( ∑ k = 1 n 1 k ( k + 1 ) = 1 − 1 n + 1 ) .Related to that, P(1) is not the quantity 1/2. It's the statement (1/2 = 1/2).The best option is to not use the abstraction of "P(n)" in actual inductive proo

True or false? If P(n) is a summation formula for the sigma sum ∑ k = 0 n f ( k ) = S ( n ) , where S(n) represents the sum in closed form, we prove the statement P(n+1) from P(n) by taking the statement P(n) and adding f(n+1) to both sides. Then we simplify S(n)+f(n+1) algebraically to show that it is S(n+1).

True

True or false? If a is a positive number and x and y are real numbers, then a x y = ( a x ) y.

True

Consider the statement "If you smoke then you will get sick". Select which one of the following statements is NOT logically equivalent to that conditional above or select You will get sick only if you smoke. Smoking it is a sufficient condition for getting sick. Being sick is a necessary condition for smoking. You are not sick therefore you did not smoke. If you are not sick, then you did not smoke. All of these statement are equivalent to "if you smoke then you will get sick".

You will get sick only if you smoke.

What is the product of two 8-bit numbers? Correct answers must be general, i.e. be valid no matter what the two 8-bit numbers are. If there is more than one correct answer, select the one with the lowest number of bits.

a 16-bit number.

What is the sum of a 16-bit and a 32-bit number? Correct answers must be general, i.e. be valid no matter what the two numbers are. If there is more than one correct answer, select the one with the lowest number of bits.

a 33-bit number.

What is the product of a 16-bit and a 32-bit number? Correct answers must be general, i.e. be valid no matter what the two numbers are. If there is more than one correct answer, select the one with the lowest number of bits.

a 48-bit number.

What is the sum of two 8-bit numbers? Correct answers must be general, i.e. be valid no matter what the two 8-bit numbers are. If there is more than one correct answer, select the one with the lowest number of bits.

a 9-bit number.

Given the following conditional statement: " if mn is odd, then n is odd and m is odd" Find the converse, inverse, contrapositive and the negation of this statement by matching the correct term with the correct statement.

converse: If n is odd and m is odd, then mn is odd Inverse: If mn is even, then n is even or m is even. contrapositive: If n is even or m is even, then mn is even. Negation: mn is odd and n is even or m is even

Check all statements that are true. f(x)=5x is of order 3x. f(x) = sin(x) is big-O of 1. If two functions are of order g, then so is their sum. f(x)=x is Ω (sqrt(x) ) If p is a polynomial of degree n, and q is a polynomial of degree m, and n=m, then p is of order q. All power functions f(x)=xⁿ, where n is a real constant, are O(eˣ). aˣ is of order bˣ exactly when a and b are equal. aˣ is O(bˣ) exactly when a<b. aˣ is Ω(bˣ) exactly when a>b. There is a "largest order", i.e. there is so

f(x)=5x is of order 3x. f(x) = sin(x) is big-O of 1. f(x)=x is Ω (sqrt(x) ) If p is a polynomial of degree n, and q is a polynomial of degree m, and n=m, then p is of order q. All power functions f(x)=xⁿ, where n is a real constant, are O(eˣ). If p is a polynomial of degree n, and q is a polynomial of degree m, and n<m, then p is O(q). If f and g are functions defined for all positive real numbers and if lim x → ∞ | f ( x ) g ( x ) | = Clim x → ∞ | f ( x ) g ( x ) | = Clim x → ∞ | f ( x ) g ( x ) | = Clim x → ∞ | f ( x ) g ( x ) | = Clim x → ∞ | f ( x ) g ( x ) | = C where C is a positive constant, then f is of order g. The triangle inequality says that for all real numbers a and b, |a + b| ≤ |a| + |b|. If two functions are O(g), then so is their sum. f(x)=x is O(x²).

For all predicates P with two variables, ∀x∃y P(x,y) → ∃y∀xP(x,y).

false

For any predicates P and Q of one variable, ∃xP(x) ∧ ∃xQ(x) → ∃x (P(x) ∧Q(x)).

false

The biconditional is the negation of the inclusive or.

false

True or false? For any binary predicate P, the statements ∀x ∃y P(x,y) and ∃y ∀x P(x,y) are logically equivalent.

false

True or false? In the context of our theory of inductive proofs, P(n) represents the quantity about which we are proving something.

false

True or false? p → ¬p is a contradiction.

false

You have a large text file of people. Each person is represented by one line in the text file. The line starts with their ID number and after that, has the person's name. The lines are sorted by ID number in ascending order. There are n lines in this file. You write a search function that returns the name of a person whose ID number is given. The simplest way to do that would be to program a loop that goes through each line and compares the ID number in the line against the given ID number. If t

log(n)

Identify which one(s) of the given expressions are tautologies. p → ( q → p ) p → ( p → q ) ( p → q ) → p ( q → p ) → p

p → ( q → p )

Identify which of the following are tautologies. p is a propositional variable. Recall that ⊕ denotes the exclusive or. p → p p ⊕ p p∨p p∧p p ↔ p

p → p p ↔ p

Select which one of the following statements is logically equivalent to p → (p→ q):

p → q

True or False? The last hex digit of an even positive integer is always 0, 2, 4, 6, 8, A, C or E.

true

True or false? "for all real numbers x, if x = 1, then 2x+3 = 2" is a proposition.

true

True or false? The two conditional statements If it is morning, then the sun rises The sun rises if it is morning have the same meaning.

true

True or false? ∃y ∀x ((2x-1)(y-3) = 0), where the domain of discourse is the set of positive integers.

true

a n = ∑ k = 1 n k 2, then a n + 1 = ∑ k = 1 n + 1 k 2.

true

True or false? ∀x (x=2 ∧ x=3 → x = 0). The domain of discourse is the real numbers.

true Since the premise x=2 and x=3 is false for all real numbers x, the conditional is true for all real numbers x.

Select all sets I for which the quantified statement ∀x ∃y (x+y <1) is true if I is both the domain for x and y. {0} is the set with just the one number 0 in it. {0,1} is the set with two numbers, 0 and 1, in it. (0,1) is the continuum of real numbers strictly between 0 and 1. [0,1] is the continuum of real numbers from 0 to 1, including 0 and 1.

{0} is the set with just the one number 0 in it. (0,1) is the continuum of real numbers strictly between 0 and 1.