CK-12-Pre-Calculus Concepts

http://www.ck12.org Chapter 16. Logic and Set Theory

Notice that there are four different regions in the Venn Diagram which correspond to the four different rows of the
conditional truth table. The only space that does not make the statement true is whenPis true (inside circleP) andQ
is false (outside circleQ).

TABLE16.10:

P Q P→Q

T T T

T F F

F T T

F F T

To negate this statement, you switch the values in the truth table and switch the shaded region in the Venn Diagram.

A different way to think about this shaded region is to see it as the space that is inPbut not inQ.
∼(P→Q)⇔P∧∼Q
Concept Problem Revisited
The statement already has several negative parts, so it is incorrect to simply switch one or both of the negations.
If I am not cold, then it is not snowing.

• P=I am cold.


• Q=It is snowing.

TABLE16.11:

P Q ∼P ∼Q ∼P→∼Q ∼(∼P→∼Q)

T T F F T F

T F F T T F

F T T F F T

F F T T T F