Math Help - define x ~ y

1. define x ~ y

If G is a group with subgroups A and B, then how do you define "x ~y" if and only if there exists a in A and b in B such that x = ayb?

2. Re: define x ~ y

that IS the definition.

i suspect what you want to do is show "~" is an equivalence relation; that is, show ~ is reflexive, symmetric and transitive.

3. Re: define x ~ y

ok so
R is an equivalence relation

Its reflexive: y ~ y
Symmetry: x ~ y then y ~ x

is it like this? i don't get the part "if and only if there exists a in A and b in B such that x = ayb"
what do i do with x = ayb

4. Re: define x ~ y

to prove that ~ is reflexive, you need to prove that for ANY x in G, you can find a in A and b in B such that x = axb.

you can't just say that x~x, because you don't KNOW that, unless you can actually produce the elements a and b.

5. Re: define x ~ y

ok got it. Thank you