00:01
For this problem, we want to show that the statement p if and only if q is logically equivalent to the statement not p if and only if not q.
00:12
And to do so we're going to use a truth table.
00:16
First off, let's set up all the possible instances for the truth value of p and q.
00:23
We have that either p is true or false.
00:29
Now q can be either true or false when p is true or true or false when p is false.
00:36
Now we can use these to construct our negations.
00:45
False, false, true, true.
00:46
And here we'll have false, true, false.
00:51
Now we have everything we need to work on our implication statements and then evaluate our final statements.
00:58
So we have p implies q.
01:01
This is going to be true when p is true and q is true, when p is false and q is true, and when p is false and q is false.
01:14
The one instance where an implication statement is false is when true implies false.
01:21
So when p is true and q is false, in this instance, p implies q is false.
01:27
So we now move to our implication statement q implies p, which will be true here...