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.

The propositions p and q are logically equivalent if

(p→q)∧(¬q→¬p)is a tautology.

Which of the following logical equivalences is a distributive law?

p∧(q∨r)⇔(p∧q)∨(p∧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.

Let P(x, y) be the statement “x has taken y”, where the universe of discourse for x is the set of all students in your class and for y is the set of all computer courses at your school. Express the proposition ∃y∀xP(x,y) in English.

There is a computer course at your school taken by all students in your class.

There is a student at your class who has taken all computer courses at your school.

There is a computer course at your school taken by at least one student in your class.

For every student in your class there is a computer course at your school taken by him (or her).

For each of the computer courses at your school there is a student at your class who has taken it.

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

∀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)∨∀xR(x,y))

∃y(∀x¬P(x,y)∨∀x¬R(x,y))

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

∃y(∀x∃zP(x,y,z)∨∃x∀zS(x,y,z))

∃y(∀x∃z¬P(x,y,z)∧∃x∀z¬S(x,y,z))

∀y(∀x∃zP(x,y,z)∧∃x∀zS(x,y,z))

∀x(∃y∃z¬P(x,y,z)∨∃y∀zS(x,y,z))

∃x∀z(∃y¬P(x,y,z)∨∃yS(x,y,z))

When the statement ∀y∃xP(x,y) is false?

There is a y such that P(x, y) is false for every x.

For every y there is an x for which P(x, y) is true.

There is an x such that P(x, y) is false for every y

There is a pair x, y for which P(x, y) is false.

For every y there is an x for which P(x, y) is false.

When the statement ∀y∀xP(x,y) is true?

There is a y for which P(x, y) is true for every x.

For every y there is an x for which P(x, y) is true.

For every x there is a y for which P(x, y) is false.

There is a pair x, y for which P(x, y) is true.

There is an x for which P(x, y) is true for every y.

Let W(x, y) mean that x has visited y, where the universe of discourse for x is the set of all students in your school and the universe of discourse for y is the set of all Web ∃x∃y∀z((x≠y)∧(W(x,z)⇔W(y,z))) sites. Express the statement by a simple English sentence.

There are two students in your school who have visited the same Web sites.

For every student in your school there exists another student who has visited only Web sites visited by the first student.

There are two students in your school such that if a Web site has been visited by the first student then this site has also been visited by the second student.

For every Web site there exist two students who have visited it.

For every Web site visited by a student in your school there exists another student who has also visited it.

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}

уф ема

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A bowl contains 8 red balls and 8 green balls. A woman selects balls at random without looking at them. How many balls must she select to be sure of having at least four balls of the same color?

Which of the following sentences is a proposition?

Let p be a proposition. The statement “It is not the case that p” is denoted by

Let p and q be propositions. The proposition that is true when both p and q are true and is false otherwise is denoted by

Let p and q be propositions. The proposition that is false when p and q are both false and is true otherwise is denoted by

Let p and q be propositions. The proposition that is true when exactly one of p and q is true and is false otherwise is denoted by

Let p and q be propositions. The proposition that is false when p is true and q is false and is true otherwise is denoted by

Let p and q be propositions. The proposition that is true when p and q have the same truth values and is false otherwise is denoted by

Find the converse of P→¬

Find the contrapositive of ¬p→q

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