sum-of-products expansions of these Boolean functions

Question

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

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

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

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

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

Answer 1

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.