Relations on the power set of $X=\{a,b,c\}$.

Question

$$R=\{(A,B) | \ \ |A|=|B|\} \ \ \ \text{and} \ \ \ S=\{(A,B)| \ \ |A|<|B|\}.$$

Are $S$ and $R$ equivalence relations ?

Can someone help me in writing a proof for this? Like I have the idea that $R$ is an equivalence relation and S is not through the general relation and set knowledge but how to give a formal proof of it?

Answer 1

To prove that something is an equivalence relation, you will need to verify the corresponding properties from the definition. A,B, and C are any elements of the given set.

1. Reflexivity: A ~ A
2. Transitivity: A ~ B and B ~ C implies A ~ C
3. Symmetry: If A ~ B then B ~ A

I will show you how to prove the second property: If A~B and B~C, then |A| = |B| and |B| = |C| by the definition of your first relation. Therefore we have |A| = |C|. By the definition of the relation the last equality means that A ~ C, which is what we needed to prove.

To prove that something is an order relation you will have to verify slightly different properties. All of them can however be verified in a similar way as shown above.