If A = {1, 2, 3, 4, 5, 6, 7}, define R on A by (x, y) epsilon R if x - y is a mu
ID: 1889542 • Letter: I
Question
If A = {1, 2, 3, 4, 5, 6, 7}, define R on A by (x, y) epsilon R if x - y is a multiple of 3. Show that R is an equivalence relation on A. Determine the equivalence classes and partition of A induced by R.Explanation / Answer
>a) Proof that R is an equivalence relation on A. Certainly this relation is reflexive: A ~ A means A u Y = A u Y and this is clearly true. To show ~ is symmetric, we need to show A ~ B -> B ~ A A ~ B means A u Y = B u Y. This implies B u Y = A u Y and this is equivalent to B ~ A. So ~ is symmetric. Finally, show ~ is transitive: A ~ B and B ~ C -> A ~ C Well, A~B means A u Y = B u Y and B~C means B u Y = C u Y Thus we have A u Y = B u Y = C u Y so A ~ C Thus ~ is an equivalence relation.