Solution to Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show … - Sikademy
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Archangel Macsika

Let X = {1,2,3,4,5,6,7} and R = {x,y/x–y is divisible by 3} in x. Show that R is an equivalence relation.

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Let X = \{1,2,3,4,5,6,7\} and R = \{(x,y)|x-y\text{ is divisible by }3\} in X. Let us show that R is an equivalence relation. Since x-x=0 is divisible by 3 for any x\in X, we conclude that (x,x)\in R for any x\in X, and hence R is a reflexive relation. If x,y\in X and (x,y)\in R, then x-y is divisible by 3. It follows that y-x=-(x-y) is also divisible by 3, and hence (y,x)\in R.

We conclude that the relation R is symmetric. If x,y,z\in X and (x,y)\in R,\ (y,z)\in R, then x-y is divisible by 3 and y-z is divisible by 3. It follows that x−z=(x−y)+(y−z) is also divisible by 3, and hence (x,z)\in R. We conclude that the relation R is transitive. Consequently, R is an equivalence relation.


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