00:01
Hi, today we are solving the question in which let us see the proof.
00:05
So here first of all let us define an equivalence relation r on the set of real numbers.
00:13
So we can take for any real numbers x and y.
00:23
So x or y belong to r if and only if x minus y is an integer.
00:37
So here now let us prove that r is an equivalence relation by showing its reflexivity, symmetry and transitivity.
00:49
So for reflexivity first of all for any real number x we need to show that x or x belongs to r.
01:01
So since x minus x is equals to 0 and 0 is an integer.
01:10
So x or x belongs to r.
01:13
So therefore r is reflexive.
01:20
Now taking for symmetric.
01:27
So for any real numbers let us denote the, let us take integer is equals to n.
01:42
So x minus y is equals to n.
01:45
So when y minus x is equals to minus is taken outside minus x minus y is equals to minus 9 which is minus n.
01:59
So minus n which is also an integer.
02:05
Since the negation of an integer is also an integer.
02:12
So therefore y or x belongs to r.
02:17
So r is symmetric.
02:19
Now checking for transitivity...