Which one of the following is not true ?
- Identity element in a group is unique.
- Inverse of an element in a group is unique
- Fourth roots of unity form an additive abelian group.
- Cancellation laws hold in a group.
The Correct Option is C
Solution and Explanation
Top Questions on Relations and functions
Let $ P_n = \alpha^n + \beta^n $, $ n \in \mathbb{N} $. If $ P_{10} = 123,\ P_9 = 76,\ P_8 = 47 $ and $ P_1 = 1 $, then the quadratic equation having roots $ \alpha $ and $ \frac{1}{\beta} $ is:
- JEE Main - 2025
- Mathematics
- Relations and functions
- Let $f: A \to B$ be defined by $f(x) = \frac{x - 2}{x - 3}$, where $A = \mathbb{R} - \{3\}$ and $B = \mathbb{R} - \{1\}$. Discuss the bijectivity of the function.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Let \( R \) be a relation defined on a set \( \mathbb{N} \) of natural numbers such that \( R = \{(x, y) : xy \text{ is a square of a natural number}, x, y \in \mathbb{N} \} \). Determine if the relation \( R \) is an equivalence relation.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- Show that the function \( f : \mathbb{R} \to \mathbb{R} \) defined by \( f(x) = 4x^3 - 5 \), \( \forall x \in \mathbb{R} \), is one-one and onto.
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
- If $f(x) = -2x^2$, then the correct statement is:
- CBSE CLASS XII - 2025
- Mathematics
- Relations and functions
Questions Asked in KCET exam
- A series LCR circuit containing an AC source of 100V has an inductor and a capacitor of reactances 24\(\Omega\) and 16\(\Omega\) respectively. If a resistance of 6\(\Omega\) is connected in series, then the potential difference across the series combination of inductor and capacitor will be
- KCET - 2025
- Electromagnetic waves
- A scientist wants to produce virus-free plant in tissue culture. Which part of the plant will he use as an explant? a) mature stem b) axillary meristem c) apical meristem d) mesophyll cell Choose the correct option from the following.
- KCET - 2025
- Tissue Culture
A wooden block of mass M lies on a rough floor. Another wooden block of the same mass is hanging from the point O through strings as shown in the figure. To achieve equilibrium, the coefficient of static friction between the block on the floor and the floor itself is
- KCET - 2025
- Newtons Laws of Motion
In an experiment to determine the figure of merit of a galvanometer by half deflection method, a student constructed the following circuit. He applied a resistance of \( 520 \, \Omega \) in \( R \). When \( K_1 \) is closed and \( K_2 \) is open, the deflection observed in the galvanometer is 20 div. When \( K_1 \) is also closed and a resistance of \( 90 \, \Omega \) is removed in \( S \), the deflection becomes 13 div. The resistance of galvanometer is nearly:
- KCET - 2025
- Viscosity
- If \( A \) is a square matrix of order \( 3 \times 3 \), \( \det A = 3 \), then the value of \( \det(3A^{-1}) \) is:
- KCET - 2025
- Matrices
Concepts Used:
Relations and functions
A relation R from a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.
A relation f from a set A to a set B is said to be a function if every element of set A has one and only one image in set B. In other words, no two distinct elements of B have the same pre-image.
Representation of Relation and Function
Relations and functions can be represented in different forms such as arrow representation, algebraic form, set-builder form, graphically, roster form, and tabular form. Define a function f: A = {1, 2, 3} → B = {1, 4, 9} such that f(1) = 1, f(2) = 4, f(3) = 9. Now, represent this function in different forms.
- Set-builder form - {(x, y): f(x) = y2, x ∈ A, y ∈ B}
- Roster form - {(1, 1), (2, 4), (3, 9)}
- Arrow Representation
