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I've already confirmed that the following expression is true with a truth table, but I need to prove this with other Boolean expressions for my assignment. The $\oplus$ symbol is exclusive or in this case.

$f \oplus (g \oplus h)=(f \oplus g) \oplus h$

I have been messing around with little expressions for a while trying to get a start on proving this, but I've not really made any progress.

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2 Answers 2

up vote 2 down vote accepted

Recall, you need "or", "not", and "and" to express the "exclusive or":

$f\oplus g$ means ($f$ or $g$) and NOT ($f$ and $g$):

$$f\oplus g \equiv (f\lor g) \land \lnot (f \land g)$$

Using $\;'\cdot\,'\,\text{ for}\; \land,\;\; '+\,'\,\text{ for}\,\,\lor\; \text{ and}\;\, \,'\, \text{ for}\;\,\lnot:$

$$f \oplus g = (f + g)\cdot(f\cdot g)'$$

Now extend that to three arguments

Expand $f\oplus(g\oplus h)$, and manipulate the expansion to obtain $(f\oplus g)\oplus h$

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You can can rewrite the $\oplus$ in terms of "not" and "or". Then, it should be easy to follow the logic.

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