Commutativity

A binary operator is commutative if changing the order of the inputs doesn’t change the output.
For example, addition (+) is commutative because a + b = b + a for any numbers a and b.
multiplication (*) is also commutative: a * b = b * a.
However, subtraction (-) and division (/) are not commutative:
- a - b is generally not equal to b - a.
- a / b is generally not equal to b / a.
- Exponentiation (^) - powers - are also not commutative: a^b is generally not equal to b^a.

Associativity

A binary operator is associative if the way in which inputs are grouped does not affect the output.
For example, addition is associative because (a + b) + c = a + (b + c) for any numbers a, b, and c.
multiplication is also associative: (a * b) * c = a * (b * c).
However, subtraction and division are not associative:
- (a - b) - c is generally not equal to a - (b - c).
- (a / b) / c is generally not equal to a / (b / c).
- Exponentiation is also not associative: (a^b)^c is generally not equal to a^(b^c).

If e*a=a*e for ALL values in the set, e is the identity.
For addition, for example, the identity is 0, because 0 + a = a + 0 = a for any number a.
For multiplication, the identity is 1, because 1 * a = a * 1 = a for any number a.

The inverse of the identity element is the identity element itself.

The inverse element is the
If a*b=b*a=e, a and b are inverse of each other.
For addition, the inverse of a number a is -a, because a + (-a) = (-a) + a = 0.
For multiplication, the inverse of a number a (assuming a \neq 0) is \frac{1}{a}, because a * \frac{1}{a} = \frac{1}{a} * a = 1.

Not all binary operators have an inverse.