Question

Which statements are logically equivalent to ~(p ^q)?

A. ~(p v q)
B. ~p -> q
C. (p ^ ~q)
D. p -> ~q
E. ~p v ~q
F. ~p ^ q

Answer 1

The statements logically equivalent to ~(p ^ q) is E. ~p v ~q. An example of a statement and its negation would be a sunny and warm weather negated to not sunny or not warm. The law of noncontradiction and the concept of a counterexample were also explained.

Step-by-step explanation:

The student's question asks which statements are logically equivalent to ~(p ^ q).

In propositional logic, the negation of a conjunction, ~(p ^ q), is equivalent to the disjunction of the negations, which is ~p v ~q (by De Morgan's law). Therefore, the correct answer is E. ~p v ~q.

Here's an example of a statement and its negation to illustrate: If the statement is "The weather is sunny and warm," the negation would be "It is not the case that the weather is sunny and warm," which is logically equivalent to "The weather is not sunny or it is not warm."

The law of noncontradiction logically implies the law of the excluded middle by stating that a proposition and its negation cannot both be true at the same time, which leads to the conclusion that for any proposition, either that proposition is true, or its negation is true (hence, no other possibilities exist).

As for counterexamples, if we consider the universal statement "All swans are white," a single observation of a black swan would be a counterexample that disproves the universal statement.