00:01
All right, so this question is a nice elementary proof.
00:03
And what we're going to do is, given the statement, let x be an integer.
00:06
If 3x plus 1 is even, then x is odd, we're going to prove that this statement is true using contraposition.
00:11
Now, contraposition refers to two statements that are organized in the following way.
00:15
This if -then statement is actually a p implies q, which is p is the statement 3x plus 1 is even, and q is the statement x is odd.
00:23
So p implies q says if p then q.
00:27
The contraposition is if not q, then not p.
00:32
So, in english words, the contradiction of this statement is if x is, excuse me, that is incorrect.
00:38
If x is even, the opposite of being odd is even.
00:42
If x is even, then 3x plus 1 is odd.
00:46
So we can go ahead and prove this statement.
00:49
Now, x is an integer, so that's all we need.
00:51
So let's go ahead and assume that, i'm going to put this in actually, actually in black end.
00:57
So let's go ahead and assume x is even.
01:02
Then if x is even, then we can write it, write x equal to two times some integer.
01:12
And k here is an integer.
01:14
This symbol means in or a part of.
01:18
If x is equal to 2k, then x is a multiple of 2...