Consider a binary operation *on N defined as \(a*b=a^3+b^3.\)Choose the correct answer.
Consider a binary operation *on N defined as \(a*b=a^3+b^3.\)Choose the correct answer.
Is * both associative and commutative?
Is * commutative but not associative?
Is * associative but not commutative?
Is * neither commutative nor associative?
The Correct Option is B
Solution and Explanation
On N, the operation * is defined as \(a*b=a^3+b^3.\)
For, a, b, ∈ N, we have: \(a*b=a^3+b^3=b^3+a^3=b*a\) [Addition is commutative in N]
Therefore, the operation * is commutative. It can be observed that:
\((1*2)*3=(1^3+2^3)*3=9*3\)= \(9^3+3^3=756\)
\(1*(2*3)=1*(2^3+3^3)=1*(8+27)=1*35=1^3+35^3=1+35^3=42876\)
∴ (1 * 2) * 3 ≠ 1 * (2 * 3) ; where 1, 2, 3 ∈ N
Therefore, the operation * is not associative.
Hence, the operation * is commutative, but not associative. Thus, the correct answer is B.
Top Questions on Relations and Functions
- Let \( R \) be a relation defined by the teacher to plan the seating arrangement of students in pairs, where \( R = \{(x, y) : x, y \text{ are Roll Numbers of students such that } y = 3x \} \). List the elements of \( R \). Is the relation \( R \) reflexive, symmetric, and transitive? Justify your answer.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
- Edge of a variable cube increases at the rate of 5 cm/s. The rate at which the surface area of the cube increases when the edge is 2 cm long is :
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
- If $f(x) = 2x^8$, then the correct statement is :
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
- If \( f(x) = -2x^8 \), then the correct statement is :
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
- If \( R \) be a relation defined as \( a \, R \, b \) iff \( |a - b|>0 \), \( a, b \in \mathbb{R} \), then \( R \) is :
- CBSE CLASS XII - 2025
- Mathematics
- Relations and Functions
Questions Asked in CBSE CLASS XII exam
Rupal, Shanu and Trisha were partners in a firm sharing profits and losses in the ratio of 4:3:1. Their Balance Sheet as at 31st March, 2024 was as follows:
(i) Trisha's share of profit was entirely taken by Shanu.
(ii) Fixed assets were found to be undervalued by Rs 2,40,000.
(iii) Stock was revalued at Rs 2,00,000.
(iv) Goodwill of the firm was valued at Rs 8,00,000 on Trisha's retirement.
(v) The total capital of the new firm was fixed at Rs 16,00,000 which was adjusted according to the new profit sharing ratio of the partners. For this necessary cash was paid off or brought in by the partners as the case may be.
Prepare Revaluation Account and Partners' Capital Accounts.- CBSE CLASS XII - 2025
- Partnership Accounts
- Explain giving three reasons why tropics show greatest levels of species diversity.
- CBSE CLASS XII - 2025
- Biodiversity
- The element of dilemma, between humanity and patriotism, elevates the character of Dr. Sadao in ‘The Enemy’. Support your answer with evidence from the text.
- CBSE CLASS XII - 2025
- Literature
- Find: \[ \frac{dy}{dx}, \quad \text{if} \quad y = x \tan x + \frac{\sqrt{x^2 + 1}}{2} \]
- CBSE CLASS XII - 2025
- Differentiation
- Solve for \( x \), \[ 2 \tan^{-1} x + \sin^{-1} \left( \frac{2x}{1 + x^2} \right) = 4\sqrt{3} \]
- CBSE CLASS XII - 2025
- Trigonometric Functions
Concepts Used:
Binary Operation
A binary operation can be understood as a function f (x, y) that applies to two elements of the same set S, such that the result will also be an element of the set S. Examples of binary operations are the addition of integers, multiplication of whole numbers, etc. A binary operation is a rule that is applied on two elements of a set and the resultant element also belongs to the same set.
Properties of Binary Operation:
- Closure Property: A binary operation * on a non-empty set P has closure property, if a ∈ P, b ∈ P ⇒ a * b ∈ P.
- Associative Property: The associative property of binary operations holds if, for a non-empty set S, we can write (a * b) *c = a*(b * c), where {a, b, c} ∈ S. Commutative Property: A binary operation * on a non-empty set S is commutative, if a * b = b * a, for all (a, b) ∈ S. Suppose addition be the binary operation and N be the set of natural numbers.
- Distributive Property: Let * and # be two binary operations defined on a non-empty set S. The binary operations are distributive if, a* (b # c) = (a * b) # (a * c), for all {a, b, c} ∈ S. Suppose * is the multiplication operation and # is the subtraction operation defined on Z (set of integers).
- Identity Element: A non-empty set P with a binary operation * is said to have an identity e ∈ P, if e*a = a*e= a, ∀ a ∈ P. Here, e is the identity element.
- Inverse Property: A non-empty set P with a binary operation * is said to have an inverse element, if a * b = b * a = e, ∀ {a, b, e}∈P. Here, a is the inverse of b, b is the inverse of a and e is the identity element.
Read More: Truth Table
Types of Binary Operation:
There are four main types of binary operations which are: