Quiz 2

Question 1

Universal quantifier, pronounced “for all”

Answer 1

∀

Question 2

Existential quantifier, pronounced “there exists”

Answer 2

∃

Question 3

Let D be the domain for x. The universal quantification of P(x) is the proposition denoted ____________ and it is defined to be true if and only if P(x) is true for every x in D.

Answer 3

∀x ε DP(x)

Question 4

(1) Theorem: If D = {x1, x2,….,xn}

Answer 4

then ∀x ε DP(x) ≡ P(x1) ∧ P(x2) ∧…∧ P(xn)

Question 5

1. To prove a proposition of the form ∀x ε DP(x) is true, P(x) must be proven true for all x in D.
2. To prove a proposition of the form ∀x ε DP(x) is false, P(x) must be proven false for at least one x in D. Such x is then called a ________

Answer 5

Counterexample

Question 6

Let D be the domain for x. The existential quantification of P(x) is the proposition denoted __________ and is defined to be true if and only if P(x) is true for at least one x in D.

Answer 6

∃x ε DP(x)

Question 7

(2) Theorem: If D = {x1, x2,….,xn}

Answer 7

then ∃x ε DP(x) ≡ P(x1) ∨ P(x2) ∨…∨ P(xn)

Question 8

1. To prove the proposition of the form ∃x ε DP(x) is true, prove that P(x) is true for at least one x in D.
2. To prove the proposition of the form ∃x ε DP(x) is false, we must prove that P(x) is false for every x in D.

Answer 8

:)