Williams offers new scholarships if and only if either Amherst reduces class size or Smith does not hire new faculty.
A) (A ∨ S) ⊃ W
B) W ≡ ∼(A ∨ S)
C) W ⊃ (A ∨ ∼S)
D) W ≡ (A ∨ ∼S)
E) (W ≡ A) ∨ ∼S

Question

Quizplus · Accepted Answer

We can break down the statement "Williams offers new scholarships if and only if either Amherst reduces class size or Smith does not hire new faculty" into two implications:
1) If Williams offers new scholarships, then either Amherst reduces class size or Smith does not hire new faculty.
2) If either Amherst reduces class size or Smith does not hire new faculty, then Williams offers new scholarships.

Implication 1 can be represented as (W ⊃ A) ∨ (W ⊃ ∼S)
Implication 2 can be represented as (A ∨ ∼S) ⊃ W

Combining these two implications using the if and only if statement, we get:

(W ⊃ A) ∨ (W ⊃ ∼S) ≡ (A ∨ ∼S) ⊃ W

Simplifying the left side of the equation, we get:

(W ⊃ A) ∨ (W ⊃ ∼S) ≡ ∼A ⊃ (W ⊃ ∼S)

Using the contrapositive of the right side, we get:

(W ⊃ A) ∨ (W ⊃ ∼S) ≡ A ∨ (A ⊃ W)

Using the tautology A ⊃ A, we can simplify the right side to just W:

(W ⊃ A) ∨ (W ⊃ ∼S) ≡ W

Therefore, the statement can be represented as W ≡ (A ∨ ∼S), which is equivalent to choice D.

Williams Offers New Scholarships If and Only If Either Amherst