GATE CS 2016 Set 1 Q06 — Boolean Algebra

The given properties of the operator $\#$ are:

• $x \# 0 = x$
• $x \# 1 = \overline{x}$
• $x \# x = 0$
• $x \# \overline{x} = 1$

These properties uniquely define the Exclusive-OR (XOR) operation, denoted by $\oplus$:

• $x \oplus 0 = x$
• $x \oplus 1 = \overline{x}$
• $x \oplus x = 0$
• $x \oplus \overline{x} = 1$

Therefore, $x \# y = x \oplus y = x\overline{y} + \overline{x}y$.

Thus, the correct option is A.