Question

Which statement correctly identifies the negation and truth table of the statement? If two polygons are congruent, then they are similar.

A) Two polygons are congruent, and they are not similar (TRUE)
B) Two polygons are congruent, and they are not similar (FALSE)
C) Two polygons are not congruent, and they are not similar (TRUE)
D) Two polygons are not congruent, and they are not similar (FALSE)

Answer 1

Choice B

Explanation:

We're interested in the negation of an implication. Those are hard to think about so let's work it out in detail.

An implication is "if P then Q". The negation of an implication is "NOT (if P then Q)" or in symbols

`not(P implies Q)`

Recall that "if P then Q" is true exactly when both P and Q are true or P is false. In symbols

`(P implies Q) equiv ((P ^^ Q) vv not P) equiv ((P vv not P) ^^ (Q vv not P)) equiv (not P vv Q)`

So the negation is

`not (P implies Q) equiv not (not P vv Q) equiv (P ^^ not Q)`

For us P is "two polygons are congruent" and Q is "two polygons are similar." So the negation of `P implies Q` is

"Two polygons are congruent and the two polygons are not similar."

The original implication was true and this statement is obviously false, consistent with a successful negation. Choice (B).