1. ## May I have a question about equivalence class?

Hi Masters.
With all due respect, could you give me a hint for the questions?

1. Let X = {1, 2, 3, 4, 5} and Y = {3, 4}. Defi ne a relation R on the power set P(X)
of X by A R B iff A union Y = B union Y
(a) Prove that R is an equivalence relation.
(b) What is the equivalence class of {1, 2}?

2. Let X be Z * Z, i.e. X is the set of all ordered pairs of the form (x, y) with x, y is an element of Z.
De fine the relation R on X as follows:
(x1, x2)R(y1, y2) iff x1^2 + x2^2 = y1^2 + y2^2
Is it an equivalence relation?

Have a great day, masters!

2. ## Re: May I have a question about equivalence class?

Originally Posted by yanirose
1. Let X = {1, 2, 3, 4, 5} and Y = {3, 4}. Define a relation R on the power set P(X)
of X by A R B iff A union Y = B union Y
(a) Prove that R is an equivalence relation.
(b) What is the equivalence class of {1, 2}?
I won't give you the complete answer. Did you check reflexivity of R? Where did you get stuck?

Originally Posted by yanirose
2. Let X be Z * Z, i.e. X is the set of all ordered pairs of the form (x, y) with x, y is an element of Z.
Define the relation R on X as follows:
(x1, x2)R(y1, y2) iff x1^2 + x2^2 = y1^2 + y2^2
Is it an equivalence relation?
More generally, let A and B be sets and let $f:A\to B$ be a function. If R is a binary relation on A defined by $xRy$ iff $f(x)=f(y)$, then R is an equivalence relation. Your question is a special case of this.

A couple of remarks about English from a non-native speaker.

1. "May I have a question about equivalence class?" Of course you may. Nobody can forbid you to have questions. Sometimes you are not allowed to ask questions, though. But this forum exists so that people can ask questions, so there is no need to ask permission.

2. "With all due respect, could you give me a hint for the questions?" Usually, "With all due respect" is followed by some phrase that may appear to challenge that respect. For example, "With all due respect, I refuse to follow this unconstitutional order!"