Questions about functions:(adsbygoogle = window.adsbygoogle || []).push({});

Let A be a set and let f: A -> A be a function. For x,y belongs to A, define x ~ y if f(x) = f(y):

a. Prove that ~ is an equivalence relation on A.

This is my guess, but I am not sure whether I'm right:

Proving reflexiveness: If (x,y) belong to A, then f(x) = f(x), therefore, (x,y) ~ (x,y).

Proving symmetry: If (x,y) belong to A, then f(x) = f(y), therefore if (y,x) belong to A, then f(y) = f(x), so (x,y) ~ (y,x).

Proving transitivity: If (x,y) and (y,z) belong to A, then if f(x) = f(y) and f(y) = f(z), then f(x) = f(z). Therefore, (x,y) ~ (x,z).

Is this right?

b. Suppose A = {1, 2, 3, 4, 5, 6} and f = {(1,2), (2,1), (3,1), (4,5), (5,6), (6,1)}. Find all equivalence classes.

I have no idea where to start with this one. Could someone start this one out? I would really appreciate it.

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# Find all equivalence classes.

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