Question

$Q, R, S$ are Boolean variables and $\oplus$ is the XOR operator. Select the CORRECT option(s).

• A. $(Q \oplus R) \oplus S = Q \oplus (R \oplus S)$
• B. $(Q \oplus R) \oplus S = 0$ when any two of the Boolean variables $(Q, R, S)$ are 0 and the third variable is 1
• C. $(Q \oplus R) \oplus S = 1$ when $Q = R = S = 1$
• D. $((Q \oplus R) \oplus (R \oplus S)) \oplus (Q \oplus S) = 1$

Hint:

- XOR has properties: commutative, associative, and $X \oplus X = 0$. - Always check with truth tables for validation.

Answer 1

The Correct Option is A, C

Step 1: XOR ($\oplus$) is associative and commutative. Hence, \[ (Q \oplus R) \oplus S = Q \oplus (R \oplus S), \] so option (A) is correct.
Step 2: For (B), consider the case: $Q=1, R=0, S=0$. Then $(Q \oplus R) \oplus S = (1 \oplus 0) \oplus 0 = 1 \oplus 0 = 1$. But the statement claims it is always 0, which is false. Hence (B) is wrong.
Step 3: For (C), if $Q=R=S=1$, then \[ (Q \oplus R) \oplus S = (1 \oplus 1) \oplus 1 = 0 \oplus 1 = 1. \] Thus, (C) is true.
Step 4: For (D), simplify: \[ (Q \oplus R) \oplus (R \oplus S) = Q \oplus S (\text{since } R \oplus R = 0). \] \[ (Q \oplus R) \oplus (R \oplus S) = Q \oplus S, \] then adding $(Q \oplus S)$ gives $(Q \oplus S) \oplus (Q \oplus S) = 0$ not 1. Hence, (D) is false. Final correct answers are (A) and (C) only.