Question

In: Computer Science

Prove algebraically ABC+A’C’D’+A’BD’+ACD=(A’+C)(A+D’)(B+C’+D)

Prove algebraically

ABC+A’C’D’+A’BD’+ACD=(A’+C)(A+D’)(B+C’+D)

Expert Solution

The above question can be solved as follows:

A B C + A’ C’ D’ + A’.B D’ + A C D = ( A’ + C ) ( A + D’ ) ( B + C’ + D )

RHS: ( A’ + C ) ( A + D’ ) ( B + C’ + D )

= A'.A + A'.D' + A.C + C.D' . ( B + C’ + D )

= 0 + A'.B.D' + A.B.C + B.C.D' + A'.C'.D' + A.C.C' + C.D.D' + A'.D'.D + A.C.D + C.D.D'

= A'.B.D' + A.B.C + B.C.D' + A'.C'.D' + 0 + 0 + 0 + A.C.D + 0

= A.B.C + A'C'D' + A'.B.D' + A.C.D + B.C.D'

= ABC(D+D') + A'C'D'(B+B') + A'.B.D'(C+C') + A.C.D (B+B') + B.C.D'(A+A')

= ABCD + ABCD' + A'C'D'B + A'C'D'B' + A'.B.D'C + A'.B.D'C' + A.C.D B' + A.C.D B+ B.C.D'A' + B.C.D'A

= ABCD + ABCD' + A'BC'D' + A'B'C'D' + A'BCD' + AB'CD

LHS:

A B C + A’ C’ D’ + A’.B D’ + A C D

= ABC(D+D') + A’C’D’(B+B') + A’BD’(C+C') + ACD(B'+B)

=ABCD+ABCD' + A’C’D’B+A’C’D’B' + A’BD’C+A’BD’C' + ACDB'+ACDB

= ABCD + ABCD' + A'BC'D' + A'B'C'D' + A'BCD' + AB'CD

= RHS

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