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Binary Operations

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Binary Operations - Lesson Summary

Let A be any arbitrary set.

A binary operation * on a set, A, is a function * : A × A → A.

* (a,b) → a * b.

In a binary operation, a pair of elements from A associates and results in a single element of A.


Example:

Q Rational Numbers

Let a = 2/3 , b = 3/4 ∈ Q

a + b = 2/3 + 3/4 = 17/12 ∈ Q

a - b = 2/3 - 3/4 = -1/12 ∈ Q

a x b = 2/3 × 3/4 = 1/2 ∈ Q

+: Q × Q Q, defined by +(a,b) = a + b

- : Q × Q Q, defined by -(a,b) = a - b

× : Q × Q Q, defined by ×(a,b) = a × b

a/b = (2/3)/(3/4) = 8/9 ∈ Q

a/b is defined only when b ≠ 0,

i.e. a/b is not defined if b = 0.

÷ : Q × Q Q, defined by ÷(a,b) = a ÷ b − → Not a Binary Operation

Also, '+', '−' and '×' are binary operations on R, but '÷' is not a binary operation on R.


Ex:

Consider A = {5, 6, 7, 8}.

Let $: A x A → A, defined by $(a,b) = max{a,b}

$(5,5) = 5, $(5,6) = 6, $(5,7) = 7, $(5,8) = 8

$(6,5) = 6, $(6,6) = 6, $(6,7) = 7, $(6,8) = 8

$(7,5) = 7, $(7,6) = 7, $(7,7) = 7, $(7,8) = 8

$(8,5) = 8, $(8,6) = 8, $(8,7) = 8, $(8,8) = 8

⇒$: A×A →A is a function, and hence, a binary operation.

If A = {a1, a2, ..., an} and '*' is a binary operation on A, then the operation table will have n rows and n columns, with the (i,j)th entry given by ai * aj.

Given an operation table with n rows and n columns, and each entry being an element of A = {a1, a2... an}, a binary operation * : A × A → A can be defined where ai * aj is the entry in the ith row and the jth column of the operation table.

The number of binary operations * : A X A → A is equal to [n(A)]n(A X A).


Example: The number of binary operations on the set {x, y} will be 24 = 16.

n(A x A) = n(A) x n(A)

'.' n(A) = 2 and n(A x A) = 4.

Q Rational Numbers

Let a = 2/3, b = 3/4 ∈ Q

a + b = 2/3 + 3/4 = 17/12

b + a = 3/4 + 2/3 = 17/12

⇒ a + b = b + a     ∀ a ∈ Q

⇒Q is commutative under addition.

⇒The binary operation + : Q × Q Q, defined by +(a, b) = a + b can also be defined

as +(a, b)= b + a.


Commutative property:

A binary operation *on set A is called commutative, if x *y = y *x, for every x, y ÎA

+ : Q × Q Q defined by +(a,b) = a + b

x : Q × Q Q defined by x(a,b) = a x b

-:  Q x Q → Q, defined by -(a,b) = a - b

÷ : Q × Q Q, defined by ÷(a,b) = a ÷ b

But a - b ≠ b - 1

a/b ≠ b/a

(1/2)-(2/3) ≠ (2/3)-(1/2)

(1/2)/(2/3) ≠ (2/3)/(1/2)

Associative property:

A binary operation * : A × A àA is said to be associative if (a * b) * c = a * (b * c), " a, b, c, ∈ A.

'.' (a + b) + c = a + (b + c), and

(a × b) × c = a × (b × c) " a, b, c ∈ Q.

+ : Q × Q Q defined by +(a,b) = a + b

x : Q × Q Q defined by x(a,b) = a x b

((2/3)-(3/4)) - (5/2) ≠ (2/3) - ((3/4) - (5/2))

((2/3)÷(3/4)) ÷ (5/2) ≠ (2/3) ÷ ((3/4) ÷ (5/2))

-: Q x Q → Q, defined by -(a,b) = a - b

÷: Q × Q Q, defined by ÷(a,b) = a ÷ b


Identity: Given a binary operation * : A × A → A, an element e ∈ A, if it exists, is called the identity for the operation *, if a * e = a = e * a, ∀ a ∈ A.

a * b = e

0 + a = a + 0 = a        ∀ a ∈ Q

If + : Q × Q Q defined by +(a,b) = a + b is a binary operation, then 0 is the identity for '+' on Q.

a × 1 = 1 × a = a ∀a ∈ Q

If X : Q × Q Q is a binary operation defined by x(a,b) = a x b, then 1 is the identity for 'x' on Q.

Inverse: Given a binary operation * : A × A → A with the identity element, e, in A, an element a ∈ A is said to be invertible with respect to the operation *, if there exists an element, b, in A such that a * b = e = b * a. Here, b is called the inverse of a and is denoted by a-1.

If + : Q × Q Q defined by +(a,b) = a + b is a binary operation, then - a is the inverse of a for the binary operation '+' on Q.

'.' a + (-a) = (-a) + a = 0, the identity for '+' on Q.

If x : Q × Q Q is a binary operation defined by x(a,b) = a x b, then 1/a is the inverse of any a ≠ 0 for the multiplication operation '×' on Q.

'.' a x 1/a = 1/a x a = 1, the identity for 'x' on Q.

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