00:01
We have to prove that if x -y is odd, then we have to prove x and y are of opposite parity.
00:14
To prove this result, we use that contrapositive method.
00:21
In contrapositive method, we will say that if p implies q, then negation of q implies negation of p.
00:30
That means we will consider negation of q and we will show negation of p.
00:35
That means here we consider that x and y are of same parity and we will show that x -y is even.
00:44
That means we have to show that, suppose this statement is p and this statement is q, then we will here show that if x and y are of same parity.
00:56
If they are of the same parity, that means they will be both even or they will be both odd, then implies x -y is even function.
01:08
We will show this contrapositively.
01:11
So for this we are considering here the proof, writing proof here.
01:17
So let us consider that x and y are of same parity.
01:24
This is the consideration we are taking and we will show that x -y is even.
01:31
So there will be the two cases, that means x and y both are even.
01:38
If they both are even, then x can be written as twice m, that is multiple of 2 and y is also written as the multiple of 2, that is twice m.
01:48
This is true for some m and n, which are belonging to the integer set z.
01:56
Now we consider here x -y, this is equal to twice m minus twice m.
02:01
This is equal to 2 into m minus m.
02:04
Therefore, x -y it is equal to 2 into x -1 and this implies that x -y is a multiple of 2...