Binary Operation

What it is

A binary operation $*$ on a set takes any two elements of that set and combines them into a third element of the same set. An identity element $e$ is one that leaves every element unchanged: $a*e = e*a = a$ for all $a$.

Worked example: finding an identity element

Find the identity element for $*$ defined by $a*b = \dfrac{ab}{4}$ for all $a,b \in \mathbb{Q}$.

Solution: Set up $a*e=a$ and solve for $e$:

Checking the other order confirms it: $e*a = \tfrac{4a}{4} = a$ too, so $e=4$ is the identity element.

Worked example: commutative vs. associative

For real numbers, $*$ is defined by $a*b = 1+ab$. Is it commutative? Is it associative?

Commutative check: $a*b = 1+ab$ and $b*a = 1+ba$. Since ordinary multiplication commutes, $ab=ba$, so $a*b = b*a$, commutative.

Associative check: compare $(a*b)*c$ against $a*(b*c)$:

These only agree when $a=c$, not for every choice of $a,b,c$, so $*$ is commutative but not associative.