Does $x(y+z)$ simplify to two variables in Boolean Algebra?

Question

Question from the title. I'm just starting with Boolean algebra and my first set of exercises contains multiple problems which simplify to a variant of this. Am I "done" these problems, or can I still simplify further?

Answer 1

In general,

A completely-specified Boolean function essentially depends on a variable if there exists an input combination such that the value of the function changes when the variable is toggled.

Obviously $x$ toggles your function when $y=z=1$; $y$ toggles the function when $x=1$ and $z=0$, and a symmetric argument can be made for $z$. Finding such combinations in general can be challenging because the problem is a version of SAT.

For the usual CNF and DNF formulas, there are some standardized ways to test/prove this. If we put your formula in DNF, both $xy$ and $xz$ are essential prime implicants; that's because they are both [trivially] prime implicants and you can immediately check that neither one implies the other.

Wikipedia [alas] does not cover the dual [and less used] notion of essential prime implicate (sometimes called essential prime implication), which is what you need to reason about the formula directly in CNF (instead of DNF which is what implicants are for), but you can find it in some books e.g. [1] [2] [3]. In your original formula, $y+z$ is a prime implicate and so is $x$, and it is even easier to see that neither is an implicate of the other, so both are essential prime implicates.