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Hi there.
00:01
In this question, r is the relation on the set of all integers by m is related to in if and only if m plus n is even.
00:08
So we have to prove that r is an equivalence relation and we have to determine the equivalence classes in the set of all integers under this relation are and we have to prove that our description is correct.
00:20
So let's see how we'll do this.
00:22
First we'll prove that r is an equivalence relation.
00:25
So for that first we'll show about the reflexivity that is.
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Is reflexive.
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So for that, let a belong to z.
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Let us consider an arbitrary element from the set of all integers.
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Let it be a.
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Then we know that a plus a is equal to 2a and this is clearly an even number.
00:47
So this implies that a is related to a for all a belong to z because our a was chosen arbitrarily.
00:56
So this is true for all integers in the set of all integers.
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So we can say that r is reflexive.
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So r is reflexion.
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Now we have to show it is symmetric.
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For symmetric, for symmetric, let a comma b belong to z and let a is related to b.
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From this we'll be getting that a plus b is e.
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Even.
01:31
So this implies that b plus a is even.
01:35
So this implies that b is related to a.
01:39
So we got that a related to b implies b related to a.
01:42
So this says that the relation r is symmetric.
01:48
R is symmetric.
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Now we have to prove about transitivity.
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Transitive.
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So for that, let a comma b comma c belong to set of all integers.
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Then let a is related to b and b is related to c.
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From here we'll be getting that a plus b is even and b plus c is even.
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So we can say that a plus b is of the form two multiplied by some integer m where m is from z and b plus c is of the form.
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2 multiplied by n where n is from set of integers.
02:37
Okay.
02:38
So here, some of two even integers is again even number.
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So a plus b plus b plus c is equal to 2 multiplied by m plus n.
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That implies a plus c plus 2b is equal to 2 times m plus n.
02:54
And then we'll be getting this implies a plus c is equal to 2 times m plus n minus 2.
03:03
B and that is 2 times m plus n minus b.
03:07
So we can see that a plus c is even because it is a multiple of 2.
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So a plus c is even.
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This implies that a is related to c.
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So we can say that a related to b and b related to c implies a related to c.
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So the relation r is transitive.
03:26
The relation r is transitive.
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So r is reflex symmetric and transitive.
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So we can say that r is an equivalence relation.
03:37
R is an equivalence, equalence relation on z, where z is the set of all integers.
03:53
So we showed or we proved the first part.
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Now we have to determine the equivalence classes of this relation or equivalence classes on z under this relation.
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So let 1 belong to z.
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Okay, we have 1 belong to z.
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Then the equivalence classes of 1, let us denote it as cl of 1, is equal to set of all a belong to z such that a is related to 1.
04:22
That is set of all a belong to z such that a plus 1 is even...