What will be the solution set of the equation $\frac{1}{x}\times0=0$?

Question

What will be the solution set of the equation $$\frac{1}{x}\times0=0$$

The problem is I am unable to decide whether we will take $0$ or not because if I take $0$ then it becomes undefined times zero which I don't know what is it equal to. Which one is correct $$(\text{Undefined})\times0=0$$ $$OR$$ $$(\text{Undefined})\times0=\text{Undefined}$$

So that these two following questions which are synonymous in the given(here) context:

$1.$ What is $(\text{Undefined})\times0$ ?

$2.$ What will be the solution set of the equation $\frac{1}{x}\times0=0$ ?

Answer 1

Undefined $\times 0$ is still undefined. The left-hand side $\frac{1}{x} \times 0$ will be $0$ whenever it makes sense, which is for all $x \neq 0$ (whatever your $x$s are supposed to be: real, integer, natural, etc). So, that is the solution set.

Answer 2

The correct equality, as you said, is $$\text{Undefined}\times 0=\text{Undefined}$$not$$\text{Undefined}\times0=0$$

In programming languages, $\text{Undefined}$ is equivalent to $\text{nan}$, such that multiplication, subtraction, addition or division by $\text{nan}$ always leads to $\text{nan}$. The solution set is then $\Bbb R-\{0\}$.

Answer 3

Undefined times anything (or plus, minus, or whatever) is undefined. Arithmetic only applies between numbers.

Even $\dfrac xx$ is undefined at zero, because you may not divide by zero. Never.

Answer 4

In each commutative ring $R$, $r\cdot 0=0$ holds for all $r\in R$, i.e., $0$ is absorbing.

To see this, $0= 0 + 0$ and so $r\cdot 0 = r\cdot(0+0) = r\cdot 0 + r\cdot 0$. Adding $-(r\cdot 0)$ on both sides gives $0=r\cdot 0$.

Thus the equation $\frac{1}{x}\cdot 0 = 0$ in the field of real numbers holds for all $\frac{1}{x}$ which are defined, i.e., $x\ne 0$.

Answer 5

As long as $x\neq0$, $x\in\mathbb{R}$.

Undefined just means it doesn't equal anything.

Also, another word you may come across is indeterminate and is different to undefined; indeterminate means it can equal almost an infinite amount of values like $\dfrac{0}{0}$.