Question

Show that no matter what relation R,S and T may be, we have R o (S o T)=(R o S) o T

Answer 1

(S o T)(x) = S(T(x)) by definition of o
Hence
R o (S o T)(x) = R(S(T(x)))
= (R o S)(T(x))
=((R o S) o T)(x)

As x is arbitrary, it must be that

R o (S o T) = (R o S) o T

Answer 2

can i do this?

Answer 3

no?

Answer 4

I see now what notation you're using.

Let me ask you, in your notation, what does R o S mean:

a(R o S)b if and only if there is a c such that aRc and cSb?

Answer 5

I'm guessing yes.

Then

a(R o (S o T))b
<=> aRc and c(S o T)b, for some c by definition of Ro(SoT)
<=> aRc and cSd and dTb, for some c and d
<=> a(R o S)d and dTb, for some d and by definition of RoS
<=> a((R o S) o T)b, by definition of (RoS)oT