Question

P and Q are two propositions. Which of the following logical expressions are equivalent?

I. ¬(P∧Q)
II. (¬P)∨(¬Q)
III. ¬(P∨Q)
a) All of I, II, III, and IV
b) I, II, and IV only
c) I, II, and III only
d) I and II

Answer 1

Expressions I and II are equivalent through De Morgan's laws, which state that the negation of a conjunction is equivalent to the disjunction of the negations, and the negation of a disjunction is equivalent to the conjunction of the negations. Expression III is not equivalent to I or II.

Step-by-step explanation:

The question is about determining which logical expressions are equivalent to each other out of the options given. Specifically, expressions I, II, III involving propositions P and Q and their negations. We can use De Morgan's laws to ascertain the equivalencies. De Morgan's laws state that the negation of a conjunction is equivalent to the disjunction of the negations, and vice versa, which can be expressed as:

• ¬(P∧Q) ≈ (¬P)∨(¬Q)
• ¬(P∨Q) ≈ (¬P)∧(¬Q)

These two equivalences show that expression I is equivalent to expression II and not to expression III.

Therefore, the correct answer is:d) I and II