In Boolean algebra, identities and simplification techniques are essential for…
In Boolean algebra, identities and simplification techniques are essential for analyzing and designing efficient digital circuits. Using appropriate laws and theorems, solve the following:
(a) Show that:
X + Y + X̄Ȳ = 1
(b) Show that:
(X + Y)·X̄Ȳ = 0
(c) Simplify the Boolean expression:
AB + A(B + C) + B(B + C)
(d) Draw the logic diagram for the expression in part (c) before and after simplification, and verify that both circuits are equivalent.
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Show answer & explanation
(a)
X + Y + X̄Ȳ
= (X + Y + X̄)(X + Y + Ȳ)
= (X + X̄ + Y)(X + Y + Ȳ)
= (1 + Y)(X + 1)
= 1·1
= 1
(b)
(X + Y)·X̄Ȳ
= (X·X̄Ȳ + Y·X̄Ȳ)
= (0·Ȳ + X̄·0)
= 0 + 0
= 0
(c)
AB + A(B + C) + B(B + C)
= AB + AB + AC + BB + BC
= AB + AC + B + BC
= AB + AC + B
= B + AC
(d)
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