Things are pleasant if, and only if, they are not too crowded. Everything too crowded is noisy. So if something isn't noisy, then something is pleasant.
-Which of the following propositions can also be proved from the premises of the above argument?
A) (∀x)Nx $ \lor $ (∀x)Px
B) (∀x)(Nx ≡ Cx)
C) (∀x)(Nx $ \lor $ Px)
D) (∀x)(Nx $ \lor $ Cx)
E) (∀x)[Nx ⊃ (∼Px • Cx)]

Question

Quizplus · Accepted Answer

The argument states that things are pleasant if they are not too crowded, which can be represented as "¬C ⇒ P" (if not crowded, then pleasant). This can be simplified using the contrapositive to "¬P ⇒ C" (if not pleasant, then crowded). This is logically equivalent to "¬C ∨ ¬P" (not crowded or not pleasant). Using De Morgan's Law, this can be rewritten as "(C ∧ P)" (crowded and pleasant) is false for everything. This means that "¬(C ∧ P)" is true for everything, which can be written as "¬C ∨ ¬P" again using De Morgan's Law. This statement is equivalent to choice C, which states that everything is either not crowded or noisy.

Things Are Pleasant If, and Only If, They Are Not ∨ \lor ∨

Things Are Pleasant If, and Only If, They Are Not $\lor$