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Let A be any arbitrary set.

A binary operation * on a set, A, is a function * : A Ã— A â†’ A.

* (a,b) â†’ a * b.

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

Example:

**Q** â†’** Rational Numbers**

Let a = 2/3 , b = 3/4 âˆˆ Q

a + b = 2/3 + 3/4 = 17/12 âˆˆ Q

a - b = 2/3 - 3/4 = -1/12 âˆˆ Q

a x b = 2/3 Ã— 3/4 = 1/2 âˆˆ Q

+: **Q** Ã— **Q** **â†’** **Q**, defined by +(a,b) = a + b

- : **Q** Ã— **Q** **â†’** **Q,** defined by -(a,b) = a - b

Ã— : **Q** Ã— **Q** **â†’** **Q**, defined by Ã—(a,b) = a Ã— b

a/b = (2/3)/(3/4) = 8/9 âˆˆ Q

a/b is defined only when b â‰ 0,

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

Ã· : **Q** Ã— **Q** **â†’** **Q,** defined by Ã·(a,b) = a Ã· b âˆ’ â†’ Not a Binary Operation

Also, '+', 'âˆ’' and 'Ã—' are binary operations on **R**, but 'Ã·' is not a binary operation on **R**.

Ex:

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

Let $: A x A â†’ 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

â‡’$: AÃ—A â†’A is a function, and hence, a binary operation.

If A = {*a*_{1}, *a _{2}*, ...,

Given an operation table with n rows and n columns, and each entry being an element of A = {*a*_{1}, *a*_{2}... *a _{n}*}, a binary operation * : A Ã— A â†’ A can be defined where a

The number of binary operations * : A X A â†’ A is equal to [n(A)]^{n(A X A)}.

**Example:** The number of binary operations on the set {x, y} will be 2^{4} = 16.

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

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

**Q** â†’** Rational Numbers**

Let a = 2/3, b = 3/4 âˆˆ Q

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

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

â‡’ a + b = b + a âˆ€ a âˆˆ Q

â‡’Q is commutative under addition.

â‡’The binary operation + : **Q** Ã— **Q** **â†’** **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* ÃŽA

+ : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b

x : **Q** Ã— **Q** **â†’** **Q** defined by x(a,b) = a x b

**-: Q x Q â†’ Q****,** defined by -(a,b) = a - b

Ã· : **Q** Ã— **Q** **â†’** **Q**, defined by Ã·(a,b) = a Ã· b

But a - b â‰ b - 1

a/b â‰ b/a

(1/2)-(2/3) â‰ (2/3)-(1/2)

(1/2)/(2/3) â‰ (2/3)/(1/2)

**Associative property:**

A binary operation * : A Ã— A Ã A is said to be associative if (a * b) * c = a * (b * c), " a, b, c, âˆˆ A.

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

(a Ã— b) Ã— c = a Ã— (b Ã— c) " a, b, c âˆˆ **Q**.

+ : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b

x : **Q** Ã— **Q** **â†’** **Q** defined by x(a,b) = a x b

((2/3)-(3/4)) - (5/2) â‰ (2/3) - ((3/4) - (5/2))

((2/3)Ã·(3/4)) Ã· (5/2) â‰ (2/3) Ã· ((3/4) Ã· (5/2))

**-: Q x Q â†’ Q**, defined by -(a,b) = a - b

Ã·: **Q** Ã— **Q** **â†’** **Q**, defined by Ã·(a,b) = a Ã· b

**Identity:** Given a binary operation * : A Ã— A â†’ A, an element e âˆˆ A, if it exists, is called the identity for the operation *, if a * e = a = e * a, âˆ€ a âˆˆ A.

a * b = e

0 + a = a + 0 = a âˆ€ a âˆˆ Q

If + : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b is a binary operation, then 0 is the identity for '+' on **Q.**

a Ã— 1 = 1 Ã— a = a âˆ€a âˆˆ Q

If X : **Q** Ã— **Q** **â†’** **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 Ã— A â†’ A with the identity element, e, in A, an element a âˆˆ 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** Ã— **Q** **â†’** **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** Ã— **Q** **â†’** **Q** is a binary operation defined by x(a,b) = a x b, then 1/a is the inverse of any a â‰ 0 for the multiplication operation 'Ã—' on **Q.**

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

Let A be any arbitrary set.

A binary operation * on a set, A, is a function * : A Ã— A â†’ A.

* (a,b) â†’ a * b.

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

Example:

**Q** â†’** Rational Numbers**

Let a = 2/3 , b = 3/4 âˆˆ Q

a + b = 2/3 + 3/4 = 17/12 âˆˆ Q

a - b = 2/3 - 3/4 = -1/12 âˆˆ Q

a x b = 2/3 Ã— 3/4 = 1/2 âˆˆ Q

+: **Q** Ã— **Q** **â†’** **Q**, defined by +(a,b) = a + b

- : **Q** Ã— **Q** **â†’** **Q,** defined by -(a,b) = a - b

Ã— : **Q** Ã— **Q** **â†’** **Q**, defined by Ã—(a,b) = a Ã— b

a/b = (2/3)/(3/4) = 8/9 âˆˆ Q

a/b is defined only when b â‰ 0,

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

Ã· : **Q** Ã— **Q** **â†’** **Q,** defined by Ã·(a,b) = a Ã· b âˆ’ â†’ Not a Binary Operation

Also, '+', 'âˆ’' and 'Ã—' are binary operations on **R**, but 'Ã·' is not a binary operation on **R**.

Ex:

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

Let $: A x A â†’ 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

â‡’$: AÃ—A â†’A is a function, and hence, a binary operation.

If A = {*a*_{1}, *a _{2}*, ...,

Given an operation table with n rows and n columns, and each entry being an element of A = {*a*_{1}, *a*_{2}... *a _{n}*}, a binary operation * : A Ã— A â†’ A can be defined where a

The number of binary operations * : A X A â†’ A is equal to [n(A)]^{n(A X A)}.

**Example:** The number of binary operations on the set {x, y} will be 2^{4} = 16.

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

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

**Q** â†’** Rational Numbers**

Let a = 2/3, b = 3/4 âˆˆ Q

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

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

â‡’ a + b = b + a âˆ€ a âˆˆ Q

â‡’Q is commutative under addition.

â‡’The binary operation + : **Q** Ã— **Q** **â†’** **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* ÃŽA

+ : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b

x : **Q** Ã— **Q** **â†’** **Q** defined by x(a,b) = a x b

**-: Q x Q â†’ Q****,** defined by -(a,b) = a - b

Ã· : **Q** Ã— **Q** **â†’** **Q**, defined by Ã·(a,b) = a Ã· b

But a - b â‰ b - 1

a/b â‰ b/a

(1/2)-(2/3) â‰ (2/3)-(1/2)

(1/2)/(2/3) â‰ (2/3)/(1/2)

**Associative property:**

A binary operation * : A Ã— A Ã A is said to be associative if (a * b) * c = a * (b * c), " a, b, c, âˆˆ A.

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

(a Ã— b) Ã— c = a Ã— (b Ã— c) " a, b, c âˆˆ **Q**.

+ : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b

x : **Q** Ã— **Q** **â†’** **Q** defined by x(a,b) = a x b

((2/3)-(3/4)) - (5/2) â‰ (2/3) - ((3/4) - (5/2))

((2/3)Ã·(3/4)) Ã· (5/2) â‰ (2/3) Ã· ((3/4) Ã· (5/2))

**-: Q x Q â†’ Q**, defined by -(a,b) = a - b

Ã·: **Q** Ã— **Q** **â†’** **Q**, defined by Ã·(a,b) = a Ã· b

**Identity:** Given a binary operation * : A Ã— A â†’ A, an element e âˆˆ A, if it exists, is called the identity for the operation *, if a * e = a = e * a, âˆ€ a âˆˆ A.

a * b = e

0 + a = a + 0 = a âˆ€ a âˆˆ Q

If + : **Q** Ã— **Q** **â†’** **Q** defined by +(a,b) = a + b is a binary operation, then 0 is the identity for '+' on **Q.**

a Ã— 1 = 1 Ã— a = a âˆ€a âˆˆ Q

If X : **Q** Ã— **Q** **â†’** **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 Ã— A â†’ A with the identity element, e, in A, an element a âˆˆ 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** Ã— **Q** **â†’** **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** Ã— **Q** **â†’** **Q** is a binary operation defined by x(a,b) = a x b, then 1/a is the inverse of any a â‰ 0 for the multiplication operation 'Ã—' on **Q.**

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