solution to equations $x(1-y)=0$ and $y(1-x)=0$

Question

I need to solve two equations:

$$ x(1-y)=0 \\ y(1-x)=0 $$

If I had one equation, ex $x(1-y)$, the solution would be $x=0$ or $y = 1$. But how do I go about solving two such equations? Are the solutions as below? $$ x = 0 \lor y=0 \\ x = 0 \lor y=1 \\ x = 1 \lor y=0 \\ x = 1 \lor y=1 $$

@MatthewLeingang I'm guessing you are referring to the equations actually not being linear, so SLE is not correct term to use. My bad, I edited the question. — Lenny White, May 16 '23 at 17:42
If $x = 0$ and $y = 1$ the first equation is satisfied while the second is not. — prcssngnr, May 16 '23 at 17:43
From a quick look $x = y$, $x = 0,1$. And yes using a logical and would be more appropriate here. — prcssngnr, May 16 '23 at 17:53
@LennyWhite Yes, I was guessing that SLE stood for “system of linear equations” but was confused by the fact that the equations weren't linear. It's an important point because nonlinear systems can't be solved in the same way that linear ones can. — Matthew Leingang, May 16 '23 at 18:06

score 2 · Accepted Answer · answered May 16 '23 at 18:12

Both equations in the system need to be satisfied.

Equation 1 says: either $x=0$ or $y=1$
Equation 2 says: either $y=0$ or $x=1$

If you choose $x=0$ from equation 1, then you can't choose $x=1$ for equation 2. So $y=0$.

Likewise, if you choose $y=1$ from equation 1, then you can't choose $y=0$ for equation 2. So $x=1$.

If you want to do it with boolean algebra, you have: $$\begin{aligned} ((x=0) \lor (y=1)) \land ((y=0) \lor (x=1)) &= ((x=0) \land (y=0)) \lor ((x=0) \land (x=1)) \lor ((y=1) \land (y=0)) \lor ((y=1) \land (x=1)) \\&= ((x=0) \land (y=0)) \lor \text{FALSE} \lor \text{FALSE} \lor ((y=1) \land (x=1)) \\&= ((x=0) \land (y=0)) \lor ((y=1) \land (x=1)) \end{aligned}$$

score 1 · Answer 2 · answered May 16 '23 at 17:47

There are many solutions, and you seem to have noticed how to find them. Just plug in numbers for $x$ (or $y$) that satisfy one of the equations, and solve the other :

If $x = 0$, then your first equation yields $0=0$, so you're left with the equation $y=0$
If $x = 1$, then your first equation yields $0=0$, so you're left with the equation $y=1$

Of course, if $(x,y)$ is neither $(0,0)$ nor $(1,1)$, the system doesn't have a solution.

So your set of solutions is $\lbrace (0,0), (1,1) \rbrace$.

score 1 · Answer 3 · answered May 16 '23 at 17:55

1

Subtracting the two equations gives $y=x$.

You have already noticed that $x=0$ or $y=1$ and therefore the solutions are $x=y=0$ or $x=y=1$.

answered May 16 '23 at 17:55

user1172706

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solution to equations $x(1-y)=0$ and $y(1-x)=0$

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