These are operations involving two operands. Addition, subtraction, multiplication and division are all binary operations. You’re practically an expert at them!
Closure
A number set is closed under an operation if the result of any two elements of the set ALWAYS produces another element in the same number set.
Example: Real numbers under multiplication
Any real number multiplied by another real number will always produce another real number thus we say that the set of real numbers is closed under multiplication.
Example: Integers under division
We can easily pick any two integers e.g. $1$ and $2$ and show that the result of the division will NOT produce an integer: $$\frac12=0.5$$
Hence the set of integers is NOT closed under division.
What do you think?
1) Is the set of real numbers closed under subtraction?
- Yes
- No
The difference of any two real numbers is always another real number.
2) Is the set of natural numbers closed under subtraction?
- No
- Yes
Using proof by counter-example: $5-9$ is $-4$ which is NOT a natural number.
Commutativity
A binary operation, $\oplus$ is commutative if the order of the operation between two elements does not matter: $$a\oplus b=b\oplus a$$
For example, multiplication is commutative: $$a\times b=b\times a$$
Subtraction is not: $$a-b\neq b-a$$
Associativity
A binary operation is associative if: $$(a\oplus b)\oplus c=a\oplus (b\oplus c)$$
Distributivity
One binary operation,$\otimes$ is distributive over another,$\odot$ if: $$a\otimes (b\odot c)=a\otimes b \odot a\otimes b$$
We know that multiplication is distributive over addition: $$a\times (b+ c)=a\times b + a\times b$$
Identity of an operation
The identity, $e$ of a binary operation, $\oplus$ is that single element which when operated on by every element $a$, produces that other element: $$a\oplus e=a$$
We already know of the additive identity: $$a+0=a$$ (any number plus $0$ is itself) and the multiplicative identity: $$a\times 1=a$$ (any number times $1$ is itself)
Inverses
This is the element which when operated on by another element produces the identity:
$$a\oplus a^{-1}=e$$