- #1
Caldus
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Questions about functions:
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.
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.