How may I define e.g. Tilde[x,y] to have certain properties?

I would like to define, e.g., the Tilde[x,y] operator to have certain properties, and then investigate whether certain expressions involving that operator are True or not. For example, if I specify that x~x for all x, and I specify that x~y implies y~x, and I specify that x~y && y~z implies x~z, is it true that x~y && x~z implies y~z? That sort of thing.

I'm not sure how to specify the properties of Tilde[x,y] and not sure how to test a statement for truth, although I guess Simplify[] might work.

Any help would be welcome,

Thanks

Paul Reiser