Re: Prove = is an equivalence relation for sets

On 3 Okt., 21:28, TheGist <theg...@xxxxxxxxxx> wrote:
For any sets A,B, C Show that '=' has the following properties
(i) Reflexive. A=A.
(ii) Symmetric. If A=B, then B=A.
(iii) Transitive. If A=B and B=C then A=C.

(i)Proof: '=' is defined to mean that two sets have the exact same
elements. Clearly A has the same elements as itself.
(ii)Proof: For all x in A and for all y in B A=B implies that
            all y are in A and all x are in B. Thus B=A as well.
(iii)Proof: For all x in A, y in B, and z in C
                A=B implies that all y are in A
             and
                all x are in B.
             B=C implies
                all z are also in B and all y are in C.
             But we know that all y are also in A since A=B.
             This A=C.

I know these are basic proofs. they hard part is getting around the fact
that they are so intuitively true.
Are my proofs correct?

You biggest obstacle is indeed one's intuitively knowing what equality
means.
Maybe you'd prefer to show that any relation ~ that is defined by
A~B :<=> forall x: P(x,A) <=> P(x,B)
for some P is an equivalence relation (in our special case P(x,A) is
just "x is an element of A").

For additional insight, assume that we'd have instead
A~B :<=> exists x: P(x,A) <=> P(x,B)
Is this necessarily an equivalence relation?

hagman
.

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