Exclusive Or Establish using only algebraic means that A B C

Exclusive Or Establish using only algebraic means that (A B) C = A (B C) and thus we are entitled to use A B C.

Solution

A B = A\'B + AB\'

A\'B + AB\' C = (A\'B + AB\')\' C + (A\'B + AB\')C\'

= ((A\'B)\' . (AB\')\')C + A\'BC\' + AB\'C\'

= (A + B\') . (A\' + B) . C + A\'BC\' + AB\'C\'

= AA\'C + ABC + A\'B\'C + BB\'C + A\'BC\' + AB\'C\'

= ABC + A\'B\'C + A\'BC\' + AB\'C\' (LHS)

B C = (B\'C + BC\')

A   (B\'C + BC\') = A\' (B\'C + BC\') + A (B\'C + BC\')\'

= A\'B\'C + A\'BC\' + A((B\'C)\' . (BC\')\')

= A\'B\'C + A\'BC\' + A((B + C\') . (B\' + C))

= A\'B\'C + A\'BC\' + ABB\' + ABC + AB\'C\' + ACC\'

= A\'B\'C + A\'BC\' + ABC + AB\'C\'

= ABC + A\'B\'C + A\'BC\' + AB\'C\' (RHS)

Therefore, both are equal.