00:01
Hello everyone, in this problem, we are given that let a be a set and we need to show that ia is the only relation on a that is both equivalence on a and also a function from a to a.
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This is equivalence equivalence relation and it is also a function from a to a.
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So here ia is the identity relation.
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Now first we show that ia is an equivalence relation.
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So for that in an identity relation i of a is equal to a for all a belongs to a.
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Let us take that to be equation 1.
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Now we need to define a is related to b by this relation.
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So i of a to be equal to b for all ab belongs to a.
01:01
So now we first prove for equivalence relation.
01:04
First we need to find that whether the given relation is reflexive.
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So for that let us take any element a which belongs to a.
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So i of a will be equal to a from the equation 1 and with this we can say that ia is reflexive.
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Now let us prove for symmetric.
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Now let us take two elements a and b which belongs to a such that a is related to b by the relation of ia.
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So now i of a will be equal to b.
01:35
So from the equation 1 we can have this value to be equal to a.
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So this implies that a is equal to b and b is also equal to a.
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So from this we can rewrite b as i of b from the equation 1.
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So now with this we can say that b is related to a by this relation.
01:55
So with this implies that the relation is symmetric.
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Now let us prove for transitive.
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So now let us take three elements abc which belongs to a such that a is related to b by the relation and again b is related to c by the relation.
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So we have this from the equation 1.
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We have the value of ia of a to be a which is equal to b and ia of b to be b which is equal to c.
02:20
Now comparing this we have this value to be a to be equal to c and we can rewrite a as ia of a which is to be equal to c from the equation 1...
