mathematics - XOR - Is it possible to get a, b, c from a⊕b, b⊕c, a⊕c?

Just got a simple question from a friend; still thinking. Let's share it!

Is it possible to get a, b, c when you have a⊕b, b⊕c, a⊕c ?

where ⊕ is the boolean XOR (exclusive OR) operator.
a,b,c are any boolean numbers with the same length.
For the sake of simplicity, consider a,b,c as bytes (length is 8 bits).

Edit: You can use other operators too.
Edit 2: Though it's preferable to use only ⊕ (if possible) (removed, use any operator)

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Note 1: Answer without explanations (e.g. yes/no) are not allowed.

Note 2: This is a puzzling programming, math-related question and could be placed on Puzzling.SE, Programming.SE, StackOverflow and Math.SE. So excuse me or migrate the question if I pasted it on the wrong SE forum.