A binary operation - Math Central

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Question from sofia, a parent:

Prove that if * is associative and commutative binary operation on a set S, then
(a*b)*(c*d) = [(d*c)*a]*b

for all a,b,c,d element in S. Assume the associative Law only for triples as in the definition that is, assume only
(x*y)*z = x*(y*z)
for all x,y,z element in S.

Hi Sofia,

On the right side let x = d*c, y = a and z = b then applying the associative law

[(d*c)*a]*b = [x*y]*z = x*[y*z] = (d*c)*(a*b)

Does that help?
Penny

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