GATE CS 2005 Q15 — Combinational Circuits

To find the output $f$, we analyze the logic gates in the given circuit from left to right:

• The upper XOR gate has inputs $x$ and $y$, so its output is: $$g_1 = x \oplus y$$
• The lower XOR gate has inputs $x$ and $z$, so its output is: $$g_2 = x \oplus z$$
• The final XOR gate combines these two intermediate outputs: $$f = g_1 \oplus g_2 = (x \oplus y) \oplus (x \oplus z)$$

Using the algebraic properties of the XOR operation (commutativity, associativity, and $A \oplus A = 0$):

Since the simplified expression for $f$ is $y \oplus z$, the output is completely independent of the input variable $x$. Therefore, $f$ is independent of $x$.