# Why is 'or' used when describing the solutions of an equation?

Why in the solutions of an equation do we use the word 'or' and not 'and'? Suppose the solutions of an equation are $-1,0,1$; we say that solutions are $-1$ or $0$ or $1$.

But the word 'or' in maths means that even one of the statements, the solutions in this case, can be true. And the word 'and' means that all the statements are true.

• For the same reason that an apple is green or red, not green and red.
– dxiv
Commented Feb 19, 2018 at 21:06

If we had this equation, for example:

$$(x - 1)(x - 2) = 0$$

We say $x$ is $1$ or $2$ because $x$ can be $1$ or $2$ to make the equation true. $x$ cannot be two values at once (e.g. $x$ is $1$ and $2$).

That said, it's perfectly valid to say: $1$ and $2$ are solutions to the equation because you're not implying that $x = 1$ and $2$. You're only implying that when $x = 1$ and when $x = 2$, the equation is true.

We don't say "the solutions of the equation are $-1$, $0$ or $1$". That's like saying "the people contributing to this MSE question and answer exchange are Rob or John". In ordinary English you'd say "Rob and John" (using "and" not as a logical connective but as a way of forming a list). So you can say "the solutions of the equation are the numbers $-1$, $0$ and $1$" or "$x$ is a solution to the equation if $x=-1$, $x = 0$ or $x = 1$".

Because even if there are multiple solutions, the unknown variable can't be more than one of them. It can't be all of them simultaneously, and therefore the solutions to $x^3-x=0$ are $x=-1,x=0$ or $x=1$.

Also, even if we disregard that impossibility, there is another reason. If we take the equation $xyz=0$, then we cannot conclude that $x=0,y=0$ and $z=0$; we do not know from that equation that all the variables are $0$. The only conclusion we can draw is that $x=0,y=0$ or $z=0$; at least one of the variables is $0$.

It depends on what is your notation. For example if we have $$f(x)=x(x-1)$$ then the roots will be $$x=0\qquad\text{or}\qquad x=1$$ because $x$ can not be $0$ and $1$ at the same time. But if you enumerate the roots like $$x_1=0\qquad\text{and}\qquad x_2=1$$ it is correct to use and.

Consider the statement 'fruits and vegetables are nutritious'.

This statement is equivalent to 'anything that is either a fruit or a vegetable is nutritious'

In logic, this would correspond to the equivalence of:

$(P \lor Q) \rightarrow R \Leftrightarrow (P \rightarrow R) \land (Q \rightarrow R)$

Thus, we see that sometimes the use of 'and' is very closely related tp he use of 'or'

I think the same thing is going here.

• It is standard English usage to use "and" to enumerate the elements of a list. This is nothing to do with the use of "and" as a logical connective and the "technical precision of logic" should be applied when translating the English language term "Jack and Jill" in statements such as "One of Jack and Jill went up the hill" and "Both Jack and Jill went up the hill" into formal logic. Commented Feb 19, 2018 at 21:26