Question

Please make notes on Binary operations

Answer 1

hi dear

In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to produce another element. More formally, a binary operation is an operation of arity two.

binary operation is a function on a set that combines two elements of the set to form a third element of the set. Examples of binary operations on the integers are addition, subtraction, multiplication,

Types of Binary Operation

Binary Addition.

Binary Subtraction.

Binary Multiplication.

Binary Division.

Here we use the formula for finding the number of binary operations with n elements n(n×n). The formula for finding the number of binary operations with n elements =n(n×n).

The binary operations are distributive if a*(b o c) = (a * b) o (a * c) or (b o c)*a = (b * a) o (c * a). Consider * to be multiplication and o be subtraction. And a = 2, b = 5, c = 4. Then, a*(b o c) = a × (b − c) = 2 × (5 − 4) = 2.Let * and o be two binary operations defined on a non-empty set A. The binary operations are distributive if a*(b o c) = (a * b) o (a * c) or (b o c)*a = (b * a) o (c * a). Consider * to be multiplication and o be subtraction. And a = 2, b = 5, c = 4. Then, a*(b o c) = a × (b − c) = 2 × (5 − 4) = 2. And (a * b) o (a * c) =  (a × b) − (a × c) = (2 × 5) − (2 × 4) = 10 − 6 = 2.

Identity

If A be the non-empty set and * be the binary operation on A. An element e is the identity element of a ∈ A, if a * e = a = e * a. If the binary operation is addition(+), e = 0 and for * is multiplication(×), e = 1.

Inverse

If a binary operation * on a set A which satisfies a * b = b * a = e, for all a, b ∈ A. a-1 is invertible if for a * b = b * a= e, a-1 = b. 1 is invertible when * is multiplication.

Solved Example for You

Question 1: Show that division is not a binary operation in N nor subtraction in N.

Answer : Let a, b ∈ N

Case 1: Binary operation * = division(÷)

–: N × N→N given by (a, b) → (a/b) ∉ N (as 5/3 ∉ N)

Case 2: Binary operation * = Subtraction(−)

–: N × N→N given by (a, b)→ a − b ∉ N (as 3 − 2 = 1 ∈ N but 2−3 = −1 ∉ N).

hopes its helps you

have a great day

take care