Finding operand from logical operation

Question

If the truth value R can be determined with this logic: $$ O_1 \oplus O_2 \oplus C_\text{in}\ = R $$

And I am given the values of $O_1, O_2, \text{and } R$, what operation can I perform to get the value of $C_\text{in}$?

Edit: After staring at a truth table for a while, am I right in thinking that: $ O_1 \oplus O_2 \oplus R = C_\text{in} $?

Answer 1

Yes.

You can derive that from the following three rules for $\oplus$:

1. $x \oplus (y \oplus z) = (x \oplus y) \oplus z$
2. $(x \oplus x) \oplus y = y$
3. $x \oplus y = y \oplus x\thinspace$

Just $\oplus (C_{in} \oplus R)$ on both sides.

Staring at a truth table is also a pretty nice way though.