# The Boolean function AB + AC is equivalent to ______.

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The Boolean function AB + AC is equivalent to ______.
1. AB + AC + BC
2. A'B'C' + ABC' + A'BC
3. ABC + A'BC + B'C'
4. ABC + ABC' + AB'C

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Correct Answer - Option 4 : ABC + ABC' + AB'C

Concept:

Important Axioms and De Morgan's laws of Boolean Algebra:

1. Double inversion $\overline{\overline A} = A$
2. A . A = A
3. A . $\overline A$  = 0
4. A + 1 = 1
5. A + A = A
6. A + $\overline A$  = 1

De Morgan's laws:

Law 1: $\overline {{\bf{A}} + {\bf{B}}} = \overline{A}\;.\overline B$

Law 2: $\overline {{\bf{A}}\;.{\bf{B}}} = \overline A +\overline B$

Calculation:

Let the given function be Y

Y = AB + AC

Now expanding by using the important properties of boolean algebra:

Y = AB(C + C̅) + AC(B + B̅)

Y = ABC + ABC̅ + ACB + ACB̅

As  ABC + ACB = ABC

Y = ABC + ABC̅ + ACB̅

Y can also be written as:

Y = ABC + ABC' + AB'C

Hence option (4) is the correct answer.

 Name AND Form OR Form Identity law 1.A=A 0+A=A Null Law 0.A=0 1+A=1 Idempotent Law A.A=A A+A=A Inverse Law AA’=0 A+A’=1 Commutative Law AB=BA A+B=B+A Associative Law (AB)C (A+B)+C = A+(B+C) Distributive Law A+BC=(A+B)(A+C) A(B+C)=AB+AC Absorption Law A(A+B)=A A+AB=A De Morgan’s Law (AB)’=A’+B’ (A+B)’=A’B’