Question

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

Answer 1

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’