- Introduction to Binary Operation
- Definition of Binary Operations
- Properties of Binary Operations
- Solved Examples

- Binary operations are mathematical operations that are performed on two operands or values. In other words, they are operations that involve two inputs or arguments. The most common binary operations are addition, subtraction, multiplication, division, and exponentiation.
- For example, in the expression 5 + 3, the plus sign (+) represents the binary operation of addition. The operands are 5 and 3, and the result of the operation is 8.
- Another example is the expression 10 / 2. The forward slash (/) represents the binary operation of division, with the operands being 10 and 2, and the result being 5.
- Binary operations are used in many areas of mathematics, including algebra, geometry, calculus, and more. They are also used extensively in computer science, where binary operations are used to manipulate and process binary data,such as numbers represented in binary code.
- In this page, we will see the general definition of binary operations in terms of functions and relations

- A binary operation * on a set A is a function * : A x A -> A
- we denote * (a,b) by a * b

Show that addition, multiplication are binary operations on N ( Set of Natural Numbers), but subtraction and division is not a binary operation on N.

for addition,we can write as

+ : N x N -> N

It is given by

+ (a,b) = a + b

Clearly a+b belongs to set of Natural Numbers

So, it is a binary operation

- : N x N -> N

It is given by

- (a,b) = a - b

Now a-b is not necessary natural numbers as 4 -5=-1

So, it is not a binary operation

x : N x N -> N

It is given by

x (a,b) = a x b

Clearly a x b belongs to set of Natural Numbers

So, it is a binary operation

÷ : N x N -> N

It is given by

÷ (a,b) = a ÷ b

Now a ÷ b is not necessary natural numbers as 1/2 is not natural numbers

So, it is not a binary operation

A binary operation ∗ on the set X is called commutative, if a ∗ b = b ∗ a, for every a, b ∈ X.

A binary operation ∗ : A x A -> A is said to be associative if (a ∗ b) ∗ c = a ∗ (b ∗ c), for all a, b, c, ∈ A

Given a binary operation ∗ : A x A -> A, an element e ∈ A, if it exists, is called identity for the operation ∗, if a ∗ e = a = e ∗ a, for all a ∈ A

Given a binary operation ∗ : A x 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 and b is called the inverse of a and is denoted by a

Find out if the binary operation ∗ : R × R → R defined by a ∗ b = a + 3b

(i) Commulative

(ii) Associative

(i) For commulative

a* b =a + 3b

b* a =b + 2a

Clearly this is not commulative

(ii) For Associative

(a ∗ b) ∗ c = a ∗ (b ∗ c)

The operation ∗ is not associative, since

(8 ∗ 5) ∗ 3 = (8 + 10) ∗ 3 = (8 + 10) + 6 = 24,

while 8 ∗ (5 ∗ 3) = 8 ∗ (5 + 6) = 8 ∗ 11 = 8 + 22 = 31

Let A= R x R and * be the binary operation on A defined by

(a,b)* ( c,d) = (a+c , b+d)

(i) Show that * is commutative and Associative

(ii) Find the identity element for * on A

(iii) Find the inverse of every element (a,b) ∈ A

(i)Commutative:

(a,b)∗(c,d)=(a+c,b+d)

(c,d)∗(a,b) =(c+a,d+b)= (a+c, b+d)

So , ∗ is commutative.

Associative:

((a,b)∗(c,d))∗(e,f)=((a+c,b+d)) ∗(e,f)

=(a+c+e,b+d+f)

(a,b)∗((c,d))∗(e,f))=(a,b)∗(c+e,d+f)

=(a+c+e,b+d+f)

Hence ∗ is associative.

(ii) Identity element:

Let (e1,e2) be the identity element

then by definition

(a+ e1) *(b + e2) = (a, b) = (e1 + a) *(e2 + b)

[(a+ e1),(b + e2)]=(a, b) = [(a + e) ,(b + e2)]

So a+ e1=a and b+e2=a

Hence e1=e2=0

So (0,0) is the identity element

(iii)Inverse:

Let (i1,i2) is inverse of element (a,b)∈ A

Now by definition.

(a+i1) * (b+i2)= 0* 0 =(i1+a) * (i2+b)

[(a+ i1),(b + i2)]=(0,0) = [(a + i1) ,(b + i2)]

So, a+ i1=0 and b+i2=0

So (-a,-b) is inverse of every elemnt (a,b) ∈ A.

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