Binary Operations

In maths, a binary operation is any operation that takes two elements from a set as input and returns another element from the same set. Binary operations are extremely common, but we simply just choose not to care about the specifics too much.

Definition §

$$\circ :S\times S\to S$$

• $\circ$ = Our binary operator. It could be anything, as long as it is defined properly
• $S$ = The set which $\circ$ operates on. For example, it could be the set of all Real Numbers.
• $\times$ = When we use the multiplication symbol with two sets, we are actually getting the Cartesian Product of them, which basically gives out another set.#tosee
• $\to$ = Maps to. It means the output is also in $S$.

#todo

Examples §

Subtraction is a binary operation on the real numbers. If we have two numbers, $a,b\in \mathbb{R}$, then the difference of two numbers, $a-b$ also gives a number in $\mathbb{R}$, no matter what number it is.

Subtraction is not a binary operation on the natural numbers. If we have two numbers, $a,b\in \mathbb{N}$, we can’t always subtract them and still end up with a number that’s in $\mathbb{N}$.#todo