Given $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$?

Question

Given that the boolean expression $x \wedge y=\mathbf{F}$, how to simplify $x \wedge \lnot y$?

Is the above question equivalent to the following question?

Find z so that $\lnot(x \wedge y)\rightarrow \left [ (x \wedge \lnot y) \Leftrightarrow z \right ]$ is a tautology.

If the two are equivalent, can you solve the problem from the two perspectives if possible?

I want formal ways which can handle two or even ten variables because this is the simplified version of my real question. I'm writing a program that has complicate logic.

    if (!a && b && d)
    {
        //do action a
    }
    else if (!a && b && !d)
    {
        //do action b
    }
    else if (!b && c || a && c)
    {
        //do action c
    }
    else if(a && !c || !b && !c)
    {
        //do action d
    }
    else
    {//do action e  }

Given !a && b && d is false, can we simplify !a && b && !d?

Given !a && b && d=!a && b && !d=false, can we simplify !b && c || a && c?

Given !a && b && d=!a && b && !d=!b && c || a && c=false, can we simplify a && !c || !b && !c?

I try to simplify the if condition in order to lower the CPU usage.

Answer 1

If we know that $x\wedge y$ is false, then $x$ and $y$ are not both true. Thus, $x\wedge\neg y$ is true if and only if [fill in the blank with one of $x,y$] is true. The filled-in blank is the desired $z$ / "simplified" version of $x\wedge\neg y$.

Added: Given that $\neg a\wedge b\wedge d$ is false, then $$\begin{align}\neg a\wedge b\wedge\neg d &= (\neg a\wedge b\wedge\neg d)\vee F\\ &= (\neg a\wedge b\wedge\neg d)\vee(\neg a\wedge b\wedge d)\\ &= (\neg a\wedge b)\wedge(d\vee\neg d)\\ &= (\neg a\wedge b)\wedge T\\ &=\neg a\wedge b.\end{align}$$

Given that $\neg a\wedge b\wedge d$ and $\neg a\wedge b\wedge\neg d$ are false (that is, that $\neg a\wedge b$ is false), then we know $$T=\neg(\neg a\wedge b)=a\vee\neg b,$$ so $$\begin{align}(\neg b\wedge c)\vee(a\wedge c) &= (a\wedge c)\vee(\neg b\wedge c)\\ &= (a\vee\neg b)\wedge c\\ &= T\wedge c\\ &= c.\end{align}$$

Given that $\neg a\wedge b\wedge d$ and $\neg a\wedge b\wedge\neg d$ and $(\neg b\wedge c)\vee(a\wedge c)$ are false (that is, that $a\vee\neg b$ is true and $c$ is false), then $$\begin{align}(\neg b\wedge\neg c)\vee(a\wedge\neg c) &= (a\wedge\neg c)\vee(\neg b\wedge\neg c)\\ &= (a\wedge T)\vee(\neg b\wedge T)\\ &= a\wedge\neg b\\ &= T.\end{align}$$

Answer 2

This is quite similar to Cameron's answer, worded a bit differently. If $x\land y$ is false, then by DeMorgan, either $x$ is false or $y$ is false.

Case 1 ($x\equiv F$). In this case $(x\land \neg y)$ is false, so $(x\land \neg y)\equiv F\equiv\ ?$.

Case 2 ($y\equiv F)$. In this case $(x\land \neg y)\equiv x\land T\equiv\ ?$

You should be able to fill in the question marks now.

Answer 3

I find a solution to solve my first set of problem, that is to use the substitution method.

Simplify $x \wedge \lnot y$: $$\begin{align} & x \wedge \lnot y \\ = & x \wedge \lnot y \vee \mathbf{F} \\ = & x \wedge \lnot y \vee x \wedge y \\ = & x \end{align}$$

Simplify !b && c || a && c, get (! a && b) || c. It seemingly isn't much simplified.

Simplify a && !c || !b && !c, get 'True'.

For the second question, I transform $\lnot(x \wedge y)\rightarrow \left [ (x \wedge \lnot y) \Leftrightarrow z \right ]$ to conjunctive normal form $(\lnot x \vee y \vee z) \wedge (x \vee \lnot z)$.

We know to make $(\lnot x \vee y \vee z) \wedge (x \vee \lnot z)$ a tautology, $(\lnot x \vee y \vee z)$ and $(x \vee \lnot z)$ should be tautology first.

From the first part, z could be x or $\lnot y$; from the second part, z could be x. As a result, the only solution is z=x.