Matching XOR Operations to Results

This solution explains the matching of Boolean Exclusive-OR (XOR) operations with their corresponding results. The key property of XOR is that it outputs 1 if the inputs are different, and 0 if they are the same.

Analyzing XOR Operation (i): $X \oplus X$

When a Boolean variable $X$ is XORed with itself, the inputs are always the same. According to the XOR truth table, if inputs are identical, the output is 0.

Therefore, $X \oplus X = 0$. This matches result (q).

Analyzing XOR Operation (ii): $X \oplus \bar{X}$

Here, a Boolean variable $X$ is XORed with its complement $\bar{X}$. These two inputs ($X$ and $\bar{X}$) are always different. If $X=0$, $\bar{X}=1$. If $X=1$, $\bar{X}=0$. Since the inputs are always different, the XOR output is always 1.

Therefore, $X \oplus \bar{X} = 1$. This matches result (p).

Analyzing XOR Operation (iii): $X \oplus 0$

XORing any variable $X$ with the constant 0 results in the variable $X$ itself. This is because XORing with 0 does not change the input's value.

If $X=0$, $0 \oplus 0 = 0$. If $X=1$, $1 \oplus 0 = 1$.

Therefore, $X \oplus 0 = X$. This matches result (s).

Analyzing XOR Operation (iv): $X \oplus 1$

XORing any variable $X$ with the constant 1 results in the inverse (complement) of $X$. This is also known as the NOT operation.

If $X=0$, $0 \oplus 1 = 1$, which is $\bar{X}$. If $X=1$, $1 \oplus 1 = 0$, which is $\bar{X}$.

Therefore, $X \oplus 1 = \bar{X}$. This matches result (r).

Final Matching

Based on the analysis, the correct matches are:

• (i) $X \oplus X$ matches (q) 0
• (ii) $X \oplus \bar{X}$ matches (p) 1
• (iii) $X \oplus 0$ matches (s) X
• (iv) $X \oplus 1$ matches (r) $\bar{X}$

This corresponds to the matching: (i) - (q), (ii) - (p), (iii) - (s), (iv) - (r).