Question

Select the Boolean function(s) equivalent to $x + yz$, where $x$, $y$, and $z$ are Boolean variables, and $+$ denotes logical OR.

• A. $x + z + xy$
• B. $(x + y)(x + z)$
• C. $x + xy + yz$
• D. $x + xz + xy$

Hint:

To check Boolean equivalence, simplify each option using identities such as $x + xy = x$ and $(x + a)(x + b) = x + ab$.

Answer 1

The Correct Option is B, C

We start with the expression: \[ x + yz. \] Check each option for equivalence: (B) \[ (x + y)(x + z) \] Using Boolean algebra: \[ (x + y)(x + z) = x + yz, \] so (B) is equivalent. (C) \[ x + xy + yz = x(1 + y) + yz = x + yz, \] because $1 + y = 1$. Thus (C) is equivalent. (A) produces unwanted term $z$ when $x=0$, $z=1$. (D) contains $xz$ which incorrectly enlarges the function. Hence the correct answers are (B) and (C). Final Answer: (B), (C)