Topics
Notes
There are several ways of understanding “if p then q” statements. We shall illustrate these ways in the context of the following statement.
r : If a number is a multiple of 9, then it is a multiple of 3.
Let p and q denote the statements
p : a number is a multiple of 9.
q : a number is a multiple of 3.
Then, if p then q is the same as the following:
1. p implies q is denoted by p ⇒ q. The symbol ⇒ stands for implies. This says that a number is a multiple of 9 implies that it is a multiple of 3.
2. p is a sufficient condition for q.
This says that knowing that a number as a multiple of 9 is sufficient to conclude that it is a multiple of 3.
3. p only if q.
This says that a number is a multiple of 9 only if it is a multiple of 3.
4. q is a necessary condition for p.
This says that when a number is a multiple of 9, it is necessarily a multiple of 3.
5. ∼q implies ∼p.
This says that if a number is not a multiple of 3, then it is not a multiple of 9.
