Mathematics BINARY OPERATIONS ON A FUNCTION

Binary Operations

We know that addition of two natural numbers is a natural number.

i.e. `a in N, b in N => a + b in N`.

Similarly `a - b in Z` if `a, b in Z`

`a xx b in Z` if `a, b in Z`

Thus there is a non-empty set `X` and an ordered pair of elements `(a, b)` of `X xx X` giving a unique element of `X` obtained by so called 'addition', 'multiplication' etc. These are called binary operations on `X`.

Let `A !=0`. A function - : `A xx A -> A` is called a binary operation. Instead of notation like `f((a, b))` or `**(a, b)`, we use the notation `a ** b` for the image of this function for `(a, b)` and call `**` a binary operation on `A`. Thus, corresponding to `(a, b) in A xx A`, a unique
element `a ** b` of `A` can be obtained by `**`.

Thus `+` is a binary operation on `N, Z, Q, R, C`.

`X` is a binary operation on `N, Z, Q, R, C`.

`-` is a binary operation on `Z, Q, R, C` as `a - b` does not necessarily belong to `N` if

`a in N, b in N`.

For example `3 in N, 7 in N`, but `3 - 7 = -4 != N`.

Similarly `+` is a binary operation on `Q - {0}, R- {0}, C - {0}`. If `b = 0, a/b` is not defined in `Q` or in `R` or in `C`.

If `a in N, b in N`, then `a/b != N` unless `b | a`.

Hence division is not a binary operation on `N .`

Commutative law

If `**` is a binary operation on set `A` and if `a ** b = b ** a, AA a, b in A`, we say `**` is a commutative operation.

`+` is commutative on `N`.

`-` is not commutative on `Z` as `a - b != b - a, a, b in Z`.

Associative law

If `**` is a binary operation on `A` and if `(a ** b) ** c, AA a, b, c in A`, we say `**` is an associative binary operation on `A`.

What is the need of this law ?

See that `(a+ b)+ c =a+ (b + c)` i.e. `+` is associative on `R`. Hence we can write `a+ b + c` without ambiguity for this expression.

`(a- b)- c != -a- (b -c) AA a, b, c in R`

Hence '-' is not associative on `R`. So we have to specify brackets while using '-' for three real numbers.

Identity Element

If `**` is a binary operation on `A` and if there exists an element `e` in `A` such that `a ** e = e ** a = a, AA a in A`, we say `e` is an identity element for `**`.

`0 + a = a + 0 = a, AA a in R`

`1 - a = a - 1 = a, AA a in R`

`:. 0` is the additive identity and `1` is the multiplicative identity in `R`.

`a - 0 !=- 0 - a` for `a in R` unless `a = 0`.

`:. ` '-' has no additive identity.

Inverse of an element

If `**` is a binary operation on `A` with an identity element `e` and if corresponding to `a in A`, there exists an element `a' in A` such that `a ** a' = a' ** a = e` where `e` is the identity element for `**` we say `a'` is an inverse of a and we denote the inverse a' of a by `a^-1`.

`:. a ** a^-1 = a^-1 ** a = e`

In `R`, every non-zero real number a has an inverse `1/a`. for multiplication.

Every element `a` has an inverse `-a` for addition in `R`.

`0` has no inverse for multiplication in `R`.