Find the implication that is false.

If 2 + 3 = 5, then pigs can fly.

If pigs can fly, then 1 + 3 = 5.

Let p and q be the propositions "It is below freezing" and "It is snowing" respectively. Express the proposition ( p ∨ q) ∧ ( p → ¬q) as an English sentence.

It is either below freezing or it is snowing, but it is not snowing if it is below freezing.

It is below freezing or snowing, but it is not snowing only if it is below freezing.

It is either below freezing or snowing, and it is not below freezing if it is snowing.

It is below freezing but not snowing.

If it is below freezing, then it is not snowing.

Let p and q be the propositions "You miss the final examination" and "You pass the course" respectively. Express the proposition ¬p ↔ q as an English sentence.

If you don't miss the final examination then you pass the course, and conversely.

That you don't miss the final examination is necessary for passing the course.

Missing the final examination is sufficient for passing the course by you.

You don't miss the final examination or you pass the course.

If you don't pass the course, you miss the final examination.

A compound proposition is a tautology if

it is always true, no matter what the truth values of the propositions that occur in it.

it is always false, no matter what the truth values of the propositions that occur in it.

it is always true whenever each of the propositions that occur in it is true.

it is only false when each of the propositions that occur in it is false.

it is always true whenever all the propositions that occur in it have the same truth values.

Which of the following logical equivalences is a distributive law?

p ∧ (q ∨ r) ⇔ ( p ∧ q) ∨ ( p ∧ r)

p ∨ (q ∨ r) ⇔ ( p ∨ q) ∨ r

p ∧ (q ∧ r) ⇔ p ∧ (q ∧ r)

A proposition is a contingency if

it is neither a tautology nor a contradiction

it is both a tautology and a contradiction

it is not logically equivalent to any tautology

Let P(x) be the statement "x spends less than three hours every weekday in class", where the universe of discourse for x is the set of students. Express the proposition " ∃x¬P(x) " in English.

There is a student who spends no less than three hours every weekday in class.

Every student doesn't spend more than three hours every weekday in class.

There is a student who doesn't spend more than three hours every weekday in class.

There is a student who spends less than two hours every weekday in class.

Every student spends no more than six hours every weekday in class.

Rewrite the statement ¬∀y (P( y) ∨ ∃x ¬S (x, y)) so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

∃y(¬P( y) ∨ ∃x ¬S (x, y))

∀y(¬P( y) ∧ ∀x S (x, y))

Rewrite the statement ¬∃y (∀xP(x, y) ∧ ∃x R(x, y)) so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

∀y(∃x¬P(x, y) ∨ ∀x ¬R(x, y))

∀y(∃x¬P(x, y) ∧ ∀xR(x, y))

∃x (∀y¬P(x, y) ∨ ∀y ¬R(x, y))

∃y (∃x ¬P(x, y) ∨ ∀x R(x, y))

∃y(∀x ¬P(x, y) ∨ ∀x ¬R(x, y))

Rewrite the statement ¬∀y (∃x∀z¬P(x, y, z) ∧ ∀x∃z ¬S(x, y, z)) so that negations appear only within predicates (that is, so that no negation is outside a quantifier or an expression involving logical connectives).

∃y(∀x∃zP(x, y, z) ∨ ∃x∀z S (x, y, z))

∃y(∀x∃z¬P(x, y, z) ∧ ∃x∀z ¬S(x, y, z))

∀y(∀x∃zP(x, y, z) ∧ ∃x∀z S (x, y, z))

∀x(∃y∃z¬P(x, y, z) ∨ ∃y∀z S (x, y, z))

∃x∀z(∃y¬P(x, y, z) ∨ ∃y S(x, y, z))

Which of the following statements is true if the universe of discourse for all variables is the set of all integers?

∀n ∀m (n > m ∨ n2 ≤ m2 )

List the members of the set {x | x is a negative integer greater than (- 8)}.

{- 7, - 6, - 5, - 4, - 3, - 2, -1}

{- 8, -7, - 6, - 5, - 4, - 3, - 2, -1}

Let A = {0, 1, 2}, B = {y, z} and C = {b, c}. Find C × B × C. ( A ∩ C) ∪ B .

{(b, y, b), (b, y, c), (b, z, b), (b, z, c), (c, y, b), (c, y, c), (c, z, b), (c, z, c)}

{(0, y, b), (1, y, b), (2, y, b), (0, z, b), (1, z, c), (2, z, c), (0, y, c), (1, y, c), (2, y, c)}

{(b, y),(b, z), (c, y), (c, z)}

{(b, 0), (b, 1), (b, 2), (c, 0), (c, 1), (c, 2), (0, y), (1, y), (2, y)}

{(b, y, c), (c, y, b), (b, z, c), (c, z, b)}

. A set A is a proper subset of a set B if

every element of A is also an element of B and A $$\ne$$ B

every element of A is also an element of B

there is an element of B that is not an element of A

there is an element of A that is also an element of B

. Let A be a set. The power set of A is

the set of all proper subsets of A

the set of all finite subsets of A

the set of all proper subsets of A and the empty set.

Let A and B be sets. The Cartesian product of A and B is

the set of all ordered pairs (a, b) where a $$\in$$ A and b $$\in$$ B

the set of all ordered pairs (b, a) where a $$\in$$ A and b $$\in$$ B

o the set of all two-element sets {a,b}where a $$\in$$ A and b $$\in$$ B

{(a,b) | (a $$\in$$ A $$\wedge$$ b $$\notin$$ B) $$\vee$$ (a $$\notin$$ A $$\wedge$$ b $$\in$$ B)}

What is the Cartesian product of A = {a, b} and B = {1, 2}?

{(a, 1), (b, 1), (a, 2), (b, 2)}

{(1, a), (1, b), (2, a), (b, b)}

{(1, 1), (2, 2), (a, a), (b, b)}

{(1, a), (a, a), (1, b), (b, b)}

{(1, 1), (a, a), (2, a), (1, b)}

Discrete Mathematics Quiz

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