Determine Boolean identities using Boolean algebra, Computer Engineering

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Prove the following identities

a. A‾B‾C‾ + A‾BC‾ + AB‾C‾ + ABC‾ = C‾
b. AB + ABC + A‾ B + AB‾C = B + AC

Ans.

a. LHS = A'B'C' + A'BC' + AB'C' + ABC'
= A'C' (B' + B) + AC' (B' + B)
= A'C' + AC' [as B'+B = 1]
= C' (A' + A)
= C' [since A'+A =1]
= RHS.
Thus Proved

b. LHS = AB + ABC + A'B + AB'C

= B + AC
= B (A + A') + AC (B + B')
= B + AC [as B + B' = A + A' = 1]
= B + AC
= RHS.
Thus Proved

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