# Extracting information from simultaneous boolean AND, XOR, and NOT

I'm looking to extract some information from a series of equations with AND, XOR and NOT. I've already covered all of the easy parts using various boolean identities, so I'm looking to now determine if there are any non-obvious sources of information.

Right now, I've got this simultaneous equation which can yield extra information where c1 and c2 are known, and a and b are unknown:

a ^ b = c1
a && b = c2

a b c1 c2
0 0 0  0
1 0 1  0
0 1 1  0
1 1 0  1


When both c1 and c2 are false, then a and b must also be false. However, I've been unable to determine any other scenarios, simultaneous or otherwise, which can yield additional information.

What I would like to know is

1. Are there any other ways of extracting information?
2. Can I prove that I have all of it?

Edit: What I'm looking for, exactly, are other simultaneous equations or identities of this, or a similar, form, which may (depending on the values of constants) allow me to prove additional values for unknowns. I have already exhausted the avenues through single-operator equations.

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It seems to me that the best you can do is:

1. If $c_2$ is true, then both $a$ and $b$ are true.
2. Otherwise, if $c_1$ is false, then both $a$ and $b$ are false.
3. Otherwise exactly one of $a$ and $b$ is true, but you don't know which.

Both logical are and logical exclusive or are symmetric functions of their two arguments. Neither one distinguishes between its two inputs, so there's clearly no way you're going to be able to distinguish the situations where one of the inputs is true and one is false.

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Oh yes, I know that, and it's fine. I'm looking for other situations, like this, which may yield information. –  DeadMG Jun 4 '12 at 2:30
In that case I don't understand your question. Maybe you could edit it to clarify what you are looking for. –  MJD Jun 4 '12 at 2:31