# Understanding the truth table of → (implies)

1. Aug 23, 2013

### MathWarrior

The truth table of it is as follows for reference:

p q p→q
T T T
T F F
F T T
F F T

I was wondering anyone can shed some light on an easier way to memorize or think about this then just memorizing the truth table. This seems to be the least intuitive logical function, to understand for me.

2. Aug 23, 2013

### johnqwertyful

"If I win the lottery (P), I will give you a million dollars (Q)."

When did I lie to you?

3. Aug 23, 2013

### Stephen Tashi

It should be easy to memorize since there is only one way to make "p implies q" false.

It will be easier to understand when you study statement-functions that have a variable and are quantifed by "for each". For example " for each number x , if x > 0 then x^3 > 0". If someone says "No! That's false. Suppose x = -1", we don't consider that such an example makes the statement false. The only way to show such a quantified statement is false is to provide an example where the "if..." part is true and the "then..." part is false.

("p implies q" has the same meaning as "if p then q".)

4. Aug 23, 2013

### Staff: Mentor

A slightly different way to look at it than Stephen Tashi described is this.

p q p→q
T T T
T F F
F T T
F F T

For the first two rows, the implication is true when both p and q are true, and the implication is false when p is true but q is false.
For the 3rd and 4th rows, when p is false, the implication is defined to be true, regardless of the value of q.

Using johnqwertyful's example, the only scenario that you would have a complaint about (i.e., that his implication is false) is when he actually does win the lottery (P is true), but he doesn't give you the million dollars (Q is false).

If he hasn't won the lottery, could give you the money or not give you the money, and the implication would still be considered true.