Logic behind bitwise operators in C

Question

I came across bitwise operations in C programming, and I realized that XOR operator can be used to swap 2 numbers in their binary bases. For example let $$i=(65)_{10}=(1000001)_{2}, \text{ and } j=(120)_{10}=(1111000)_{2}$$.

Let $\oplus$ be the XOR operator, then observe that if I started with any one of them, say $i$ and following the following procedure:

1)replace its value with the $\oplus$ value, yielding $$i=(0111001)_{2},j=(1111000)_{2}$$

2) replace the other variable($j$) with another $\oplus$ value derived from the new $i$ and old $j$, yielding $$i=(0111001)_{2},j=(1000001)_{2}$$

3)replace the original variable $i$ with $\oplus$ value again, yielding $$i=(1111000)_{2},j=(1000001)_{2}$$

which shows that we would somehow have their values swapped. I found this way of programming online and I definitely can’t understand how people think of the logic aspect of this. I would think it’s linked to the truth table as follows, which shows by division of cases that the values can be swapped.

However, I am still uncertain about the full reasoning why this works, like whether there is any mathematical theorems that I should know that can aid me in my understanding.

PS: Sorry if the question is off-topic here, it feels like a programming question, but I feel that I more concerned about the “logic” rather than the programming. I also drew the table myself on MS word since I can't get the latex one to work somehow.

Answer 1

Note that you can do the same thing without bitwise operators (at least for unsigned integer types since they can't overflow into undefined behavior):

        // i == x     j == y
i += j; // i == x+y   j == y
j -= i; // i == x+y   j == -x
i += j; // i == y     j == -x
j = -j; // i == y     j == x

Now if we do this bit for bit, but modulo 2 instead of modulo UINT_MAX+1, the XOR operation implements both addition and subtraction, and the final negation is a no-op because $-1\equiv 1$ and $-0\equiv 0 \pmod 2$. So what is left in the bitwise version is exactly

i ^= j; j ^= i; i ^= j;

Answer 2

In algebraic terms, the XOR operator (or $\oplus$) is nothing other than addition modulo $2$: use $1$ and $0$ for true and false, along with $1 \oplus 1 = 0$.

Now, since addition modulo $2$ is associative and commutative, and both elements are their own inverses, we have $$\begin{align} d &= b \oplus c\\ &= b \oplus (a \oplus b)\\ &= b \oplus (b \oplus a)\\ &= (b \oplus b) \oplus a\\ &= a.\\ \end{align}$$

We can show $e = b$ using similar reasoning.

Answer 3

You already answered your question, but if you want an algebraic explanation note that for any $x$:

$$x \oplus 0 = x$$

$$x \oplus x = 0$$

So:

$$i_0 = i, j_0 = j$$

$$i_1 = i_0 \oplus j_0, j_1 = j_0$$

$$i_2 = i_1, j_2 = i_1 \oplus j_1 = i_0 \oplus j_0 \oplus j_0 = i_0$$

$$i_3 = i_2 \oplus j_2 = i_1 \oplus i_0 = i_0 \oplus j_0 \oplus i_0 = j_0, j_3 = j_2 = i_0$$