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If $A$ and $B$ are the inputs to a half adder, a half subtractor. $X$ and $Y$ are the Sum and Difference and $\mathrm{C}$ is a Carry of a Half adder and $D$ is a borrow of a Half subtractor.
$X \oplus Y$ and $C \odot D$ respectively

  1. $0, B$
  2. $1, \mathrm{B}$
  3. $1, B^{\prime}$
  4. $0, B^{\prime}$
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for Half Adder $\Rightarrow$ Sum $X=A$ xor $B, \quad$ Carry $C=A B$
for half Subtractor $\Rightarrow$ Difference $Y=A$ xor $B$, Borrow $D=A ' B$
So, $(X$ xor $Y)=0$ and $(C$ xnor $D)=B^{\prime}$
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