understanding analysis 2nd edition. Exercise 1.5.5 (a.) Why is A~A for every set

Question

understanding analysis 2nd edition. Exercise 1.5.5
(a.) Why is A~A for every set A (I am wondering on what "~" means in this case. I've learned it can have a variety of meanings)

(b.) Given sets A and B, explain why A~B is equivalent to asserting B~A. (what is this saying in the first place ?)

(c.) For three sets A,B, and C, show that A~B and B~C implies A~C. These three properties are what is meant by saying that ~ is an equivalence relation. (What am I suppose to do here, am I suppose to talk about symmetric, transitive, and reflexive in all of these.)

-Beginning proofs student. 1st semester of stuff.

Explanation / Answer

Here the symbol stands for equivalence relation.

A is equivalent to itself.

Hence A~A

b) If A~B, since equivalence relation is commutative, it follows that B~A

Hence A~B is equivalent to B~A

c) This is called transitive property of equivalence relations.

If A~B, and B~C then it follows that A~C

(Equivalence relation is a relation which is reflexive, transitive and symmetric.)

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