Question

The Boolean function $AB + AC$ is equivalent to:

• A. $AB + AC + BC$
• B. $A'BC' + ABC' + A'BC$
• C. $ABC + A'BC + B'C'$
• D. $ABC + ABC' + AB'C$

Hint:

Canonical expansion always ensures every term contains all variables using $(X + X')$ multiplication.

Answer 1

The Correct Option is D

Concept: To convert a Boolean expression into canonical Sum of Products form, we expand missing variables using the identity $X + X' = 1$ so that each term contains all variables.

Step 1: Expand first term \[ AB = AB(C + C') \] \[ = ABC + ABC' \]

Step 2: Expand second term \[ AC = AC(B + B') \] \[ = ABC + AB'C \]

Step 3: Combine terms \[ AB + AC = ABC + ABC' + ABC + AB'C \] Using idempotent law ($X + X = X$): \[ = ABC + ABC' + AB'C \]

Step 4: Interpretation The final expression is a canonical SOP form containing all minterms covered by the original expression. Final Answer: \[ \boxed{ABC + ABC' + AB'C} \]