Exclusive OR / X-OR Gate

The Exclusive-OR (X-OR) gate is a two-input, one-output gate whose defining behaviour is that its output goes HIGH (1) only when its two inputs are at genuinely DIFFERENT logic levels from each other -- one 0 and the other 1, in either order -- and stays LOW (0) whenever the two inputs AGREE (both 0, or both 1). Put another way, an X-OR gate's output is high whenever an ODD number of its inputs are high (here, exactly one out of two). Its Boolean expression is $C=A\oplus B$, and can equivalently be written out in full sum-of-products form as $C=A\overline{B}+\overline{A}B$ (Fig. 16.32) -- capturing exactly the two input rows (A=1,B=0 and A=0,B=1) where the output is high. Because it effectively COMPARES its two inputs and signals whether they differ, the X-OR gate is an extremely useful building block in computational and arith …