Boolean simplification, 5 variables

I'm currently learning for my maths exam, and in the part about boolean algebra I came across an exercise that I can't seem to solve. I probably only need the first few steps to get started.

$$(xyz + uv)(x+\overline{y}+\overline{z}+uv)$$

Usually, if I get into trouble, I can fall back to a truth table or VK-diagram, but that's just too much work for 5 variables.

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Multiply the terms in the two brackets. You get: $$xyz + 0 + 0 + xyzuv + xuv + \overline{y}uv + \overline{z}uv + uv$$ $$xyz(1+uv) + uv(1+x+\overline{y}+\overline{z})$$ $$xyz + uv$$

NOTE:

$1 + x = 1$

$1.x = x$

$x.x = x$

$x.\overline{x} = 0$

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Wow, it really seems simple this way. Exactly what I was looking for! Thanks! – Simon Verbeke Jun 6 '13 at 12:23

Let's write it like this: \begin{align} (xyz + uv)((x+\bar y + \bar z) + uv)&\equiv((xyz)(x+\bar y + \bar z) + (xyz)(uv))+(uv(x + \bar y + \bar z) + uvuv) \\ &\equiv ((xyz)(x+\bar y + \bar z) + xyzuv)+(uv(x + \bar y + \bar z) + uv)\\ &\equiv ((xyz)(x+\bar y + \bar z) + xyzuv)+(uv) \\ &\equiv ((xyzx + xyz\bar y + xyz\bar z) + xyzuv)+(uv) \\ &\equiv ((xyz + 0 + 0) + xyzuv)+(uv) \\ &\equiv (xyz + xyzuv)+(uv) \\ &\equiv (xyz)+(uv) \\ \end{align}

From line 1 to line 2, we distributed. From 2 to 3, we noted that $ab + a \equiv a$. From 3 to 4, we distributed again. From 4 to 5, we noted that $a\bar a\equiv 0$ (contradiction), which we remove from 5 to 6. 6 to 7 we note $ab + a \equiv a$.

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Is $a\bar a\equiv c$ the same as $a\overline{a} = 0$ ? I've never seen the first notation, we use the second. – Simon Verbeke Jun 6 '13 at 12:32
@SimonVerbeke In the class I took, we used $c$ to denote a contradiction, and $\tau$ to denote a tautology. We also used the $a\land b$ and $a \lor b$ operators instead of $ab$ and $a+b$, respectively. I'll edit my question to use the $0$ to denote contradiction. – apnorton Jun 6 '13 at 12:35
Ok, thank you! Yours is a good solution too, but I find lsp's solution simpler. So I'm going to accept his as the answer. – Simon Verbeke Jun 6 '13 at 12:37