Exercise 21 Page 71

Let's consider each of the statements one at a time using the given p and q. p =& It does not snow q =& I will run outside

Conditional Statement

We can write the conditional statement, p→ q, in an if-then form. If it does not snow, then I will run outside.This is a false statement, as not snowing is not the only criteria for you running outside or not. You might be grounded, so even though it is not snowing you are not allowed to run out.

Converse

The converse of a conditional statement, q→ p, exchanges the hypothesis and the conclusion of the conditional statement. If I will run outside, then it does not snow. Again, it does not logically follow that if you run outside, it does not snow. What if you want to build a snowman or make a snow angel? Then you would run out when it snows.

Inverse

The inverse of a conditional statement, ~ p→ ~ q, requires us to negate the hypothesis and the conclusion of the conditional statement. Note that the hypothesis already contains a not. Therefore, when we negate the hypothesis we get a double negation, which is the same thing as erasing the original not. If it snows, then I will not run outside. By the same logic we know the conditional and converse statements are false, we also know that the inverse is false.

Contrapositive

The contrapositive of a conditional statement, ~ q→ ~ p, starts out with the converse of the conditional statement. Then we have to negate the hypothesis and the conclusion. Again, notice that negating the original hypothesis results in a double negation. If I do not run outside, then it snows. Again, using the same logic as before, we know the contrapositive is false.