   Chapter 1.4, Problem 14E

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# Assume that ∗ is a binary operation on a non empty set A , and suppose that ∗ is both commutative and associative. Use the definitions of the commutative and associative properties to show that [ ( a ∗ b ) ∗ c ] ∗ d = ( d ∗ c ) ∗ ( a ∗ b ) for all a , b , c and d in A .

To determine

To prove: [(ab)c]d=(dc)(ab) for all a,b,c, and d in A, where is a binary operation which is both associative and commutative on the non empty set A.

Explanation

Given Information:

An associative and commutative binary operation on the non empty set A is *.

Formula Used:

If is an associative binary operation on the non empty set A then for all x,y,zA, we have, x(yz)=(xy)z.

If is an commutative binary operation on the non empty set A then for all x,yA, we have, xy=yx.

Explanation:

Consider the left side of,

[(ab)c]d=(dc)(ab)

By commutative property,

[(ab)c]d=[c(ab)]d

By associative property,

[c(ab)]d

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