I'm given this expression$$ (x+y'+z')(x'+z') $$the $'$ meaning not. I have to simplify this to 3 literals and show my answer as a product of sums.
Every calculator I check says the answer is $(x'y')+z'$. So far all I can think of to do as the first step is to expand the given expression using distribution giving me$$ xx'+xz'+x'y'+y'z'+x'z'+z'z'. $$From there I know $xx'$ is $1$ and $z'z'$ is $z'$, giving me $xz'+x'y'+y'z'+x'z'+z'$ and this is where I get stuck. Any suggestions?
$\endgroup$ 72 Answers
$\begingroup$Expression $xx'$ means “$x$ is true and not $x$ is true”, so $xx'=0$. Thus the expression is:$$ xz'+x'y'+y'z'+x'z'+z'z'=(x+x')z'+x'y'+y'z'+z'= x'y' + (y'+1)z'=x'y'+z' $$
$\endgroup$ $\begingroup$A useful principle is:
Reduction
$x(x'+z) = x + z$ (given $x$, the $x'+z$ term reduces to just $z$)
So:
$$(x+y'+z')(x'+z') = (x+y')x'+z'=y'x'+z'$$
$\endgroup$
