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Find the sum-of-products expansions of these Boolean functions.

$a)$ $F(x, y) = \text{~}x + y$

$b)$ $F(x, y) = x \text{~}y$

$c)$ $F(x, y) = 1$

$d)$ $F(x, y) = \text{~}y$

Shobhit
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Reem
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1 Answers1

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Each of the given boolean functions is already expressed as a sum of products.

For example, the first can be thought of as $$F(x, y) = \lnot x + y = \lnot x\cdot 1 + y \cdot 1$$

The second: $$F(x, y) = x(\lnot y) + 0$$

If you had an expression, say $F(x, y) = (\lnot x + \lnot y)(x + y)$, then we have a product of sums, so we'd need to distribute to get $$\begin{align} F(x,y) & = \lnot x(x + y) + \lnot y(x + y) \\ \\ & = \lnot xx + \lnot x y + \lnot y x + \lnot y y \\ \\ & = 0 + \lnot xy +\lnot y x + 0 \\ \\ &= \lnot xy + \lnot y x\end{align}$$ which is now a sum of products.

amWhy
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  • Reem, you need to follow up with questions if you don't understand an answer. Also, in future posts, they are better received on the site, in general, when you share your thoughts, or even simply explain your confusion in your post. I'll follow up with comments, as needed, when I answer. But if you are finding answers helpful, be sure to accept one for any given question! ;-) – amWhy Sep 28 '13 at 14:29