The bitwise $\mathrm{OR}$ of integers $A$ and $B$, $A\ \mathrm{OR}\ B$, is defined as follows:

• When $A\ \mathrm{OR}\ B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if at least one of $A$ and $B$ is $1$, and $0$ otherwise.

For example, we have $3\ \mathrm{OR}\ 5 = 7$ (in base two: $011\ \mathrm{OR}\ 101 = 111$).
Generally, the bitwise $\mathrm{OR}$ of $k$ integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1\ \mathrm{OR}\ p_2)\ \mathrm{OR}\ p_3)\ \mathrm{OR}\ \dots\ \mathrm{OR}\ p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots p_k$.

The bitwise $\mathrm{XOR}$ of integers $A$ and $B$, $A\ \mathrm{XOR}\ B$, is defined as follows:

• When $A\ \mathrm{XOR}\ B$ is written in base two, the digit in the $2^k$'s place ($k \geq 0$) is $1$ if exactly one of $A$ and $B$ is $1$, and $0$ otherwise.

For example, we have $3\ \mathrm{XOR}\ 5 = 6$ (in base two: $011\ \mathrm{XOR}\ 101 = 110$).
Generally, the bitwise $\mathrm{XOR}$ of $k$ integers $p_1, p_2, p_3, \dots, p_k$ is defined as $(\dots ((p_1\ \mathrm{XOR}\ p_2)\ \mathrm{XOR}\ p_3)\ \mathrm{XOR}\ \dots\ \mathrm{XOR}\ p_k)$. We can prove that this value does not depend on the order of $p_1, p_2, p_3, \dots p_k$.