By Edward A. Bender
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Necessary and sufficient: Combining the two previous bulleted items, we see that “p is necessary and sufficient for q” is equivalent to p ⇔ q, the biconditional. Notice that we simply combined “necessary” and “sufficient”, just as we combined “if” and “only if” earlier to get the biconditional. ” To say “p unless q” is, formally, to specify the statement form ∼q ⇒ p. ” Think of a night watchman sitting in his office with the fire alarm on the wall. Since the alarm isn’t ringing he relaxes, maybe even takes a nap.
Why is this? For each particular choice of x and y, the predicates become statement variables and so we are back in propositional logic. What happens when quantifiers are involved? Theorem 2 tells us how to move ∼ through quantifiers. Sometimes we can move quantifiers through ∨ and ∧, and sometimes not: True: False: False: True: ∀ x ∈ D, ∃ x ∈ D, ∀ x ∈ D, ∃ x ∈ D, (P (x) ∧ Q(x)) (P (x) ∧ Q(x)) (P (x) ∨ Q(x)) (P (x) ∨ Q(x)) ⇔ ⇔ ⇔ ⇔ (∀ x ∈ D, (∃ x ∈ D, (∀ x ∈ D, (∃ x ∈ D, P (x)) ∧ (∀ x ∈ D, P (x)) ∧ (∃ x ∈ D, P (x)) ∨ (∀ x ∈ D, P (x)) ∨ (∃ x ∈ D, Q(x)) Q(x)) Q(x)) Q(x)) In the exercises, you will be asked to explain this.
The statement “ ∃x ∈ D, S(x)” is true if and only if S(x) is true for at least one x ∈ D; otherwise the statement “ ∃x ∈ D, S(x)” is false. If the value of D is clear, we may write simply ∃x S(x). In terms of truth sets: • “ ∀x ∈ D, S(x)” is equivalent to saying that the truth set of S(x) contains the set D. • “ ∃x ∈ D, S(x)” is equivalent to saying that the truth set of S(x) contains at least one element of the set D One can view much of mathematics as an attempt to understand the truth sets of certain predicates.
A Short Course in Discrete Mathematics by Edward A. Bender