EXAMPLE 2.2.11 EXAMPLE 2.2.11

1. Select the statement that is logically equivalent to "If today is Sunday, then school is closed."

A. If today isn't Sunday, then school isn't closed.

B. If school is closed, then today is Sunday.

C. If school isn't closed, then today isn't Sunday.

D. A, B, & C are all equivalent to the statement above.

 

SOLUTION
Let p be the statement "Today is Sunday." Let q be the statement "School is closed." The statement "If p, then q" is equivalent to its contrapositive, "If not p, then q."
In words, the equivalent statement is "If school isn't closed, then today isn't Sunday," choice C.
The given statement is NOT equivalent to its converse (choice B) and it is NOT equivalent to its inverse (choice A).
 

2. Select the statement that is logically equivalent to "If you are a duck, then you aren't willing to waltz." (Adapted from Lewis Carrol.)

A. If you aren't willing to waltz, then you are a duck.

B. If you aren't a duck, then you are willing to waltz.

C. If you are willing to waltz, then you aren't a duck.

D. A, B & C are all equivalent to the given statement.

 

SOLUTION
Let p be the statement "You are a duck." Let q be the statement "You aren't willing to waltz." The statement "If p, then q" is equivalent to its contrapositive, "If not q, then not p." This is choice C.
The given statement is NOT equivalent to its converse (choice A) and it is NOT equivalent to its inverse (choice B).
 

 

3. Select the statement that is not equivalent to "If I don't invest wisely, then I'll lose my money."

A. I invest wisely or I lose my money.

B. If I don't lose my money, then I invested wisely.

C. I lose my money or I invest wisely.

D. If I invest wisely, then I won't lose my money.

SOLUTION
Let p be the statement "I don't invest wisely." Let q be the statement "I don't lose my money." The statement "If p, then q" is NOT equivalent to its INVERSE (choice D). The correct choice (the non-equivalent statement) is D.
The statement "If p, then q" IS equivalent to its contrapositive (choice B), and is also equivalent to "not p, or q" (choice A) which in turn is equivalent to "q, or not p" (choice C).