Question

Let \(\mathrm{X}=\\{\mathrm{a}, \mathrm{b}, \mathrm{c}, \mathrm{d}\\}\) then one-one mapping \(\mathrm{f}: \mathrm{X} \rightarrow \mathrm{X}\) such that \(\mathrm{f}(\mathrm{a})=\mathrm{a}, \mathrm{f}(\mathrm{b}) \neq \mathrm{b}, \mathrm{f}(\mathrm{d}) \neq \mathrm{d}\) are given by (i) \(\\{(\mathrm{a}, \mathrm{a}),(\mathrm{b}, \mathrm{c}),(\mathrm{c}, \mathrm{d}),(\mathrm{d}, \mathrm{b})\\}\) (ii) \(\\{(\mathrm{a}, \mathrm{a},),(\mathrm{b}, \mathrm{d}),(\mathrm{c}, \mathrm{c}),(\mathrm{d}, \mathrm{b})\), (iii) \(\\{(\mathrm{a}, \mathrm{a}),(\mathrm{b}, \mathrm{d}),(\mathrm{c}, \mathrm{b}),(\mathrm{d}, \mathrm{c})\\}\) (a) only (i) is true (b) (i) \& (ii) is true (c) (i), (ii) and (iii) are true (d) only (iii) is true

Short answer

The correct answer is (c) (i), (ii) and (iii) are true.

Step-by-step solution

Analyze Mapping (i)

The given mapping is: \[ \{ (a, a), (b, c), (c, d), (d, b) \} \] Now, check whether this mapping satisfies the given conditions. 1. f(a) = a: This condition is satisfied. 2. f(b) ≠ b: This condition is also satisfied since f(b) = c. 3. f(d) ≠ d: This condition is satisfied since f(d) = b. Since all the conditions are met, Mapping (i) is a valid one-to-one mapping.

Analyze Mapping (ii)

The given mapping is: \[ \{ (a, a), (b, d), (c, c), (d, b) \} \] Now, check whether this mapping satisfies the given conditions. 1. f(a) = a: This condition is satisfied. 2. f(b) ≠ b: This condition is also satisfied since f(b) = d. 3. f(d) ≠ d: This condition is satisfied since f(d) = b. Since all the conditions are met, Mapping (ii) is also a valid one-to-one mapping.

Analyze Mapping (iii)

The given mapping is: \[ \{ (a, a), (b, d), (c, b), (d, c) \} \] Now, check whether this mapping satisfies the given conditions. 1. f(a) = a: This condition is satisfied. 2. f(b) ≠ b: This condition is also satisfied since f(b) = d. 3. f(d) ≠ d: This condition is satisfied since f(d) = c. Since all the conditions are met, Mapping (iii) is also a valid one-to-one mapping.

Choose the Correct Option

We have determined that each of the given mappings (i), (ii), and (iii) satisfy all the given conditions for a valid one-to-one mapping. Therefore, the correct answer is (c) (i), (ii) and (iii) are true.

Key concepts

Understanding Functions

In mathematics, a function is a relation between a set of inputs and a set of permissible outputs. Each input, often called an "independent variable," is related to exactly one output. This one-to-one correspondence is fundamental for understanding how functions operate.
Functions serve as a way to map one set to another. Let's consider a few basic ideas to cement this concept:

• A function assigns each element in the domain (input) to a single element in the codomain (output).
• The notation \( f: X \rightarrow Y \) conveys that function \( f \) maps elements from set \( X \) to set \( Y \).
• A one-to-one function ensures that no two different inputs map to the same output.

Understanding functions helps in visualizing and creating mappings like those in the exercise, where each element from set \( X \) is matched in specific ways.

Exploring Set Theory

Set theory is the study of collections known as sets, which are used to group objects together. It is foundational to mathematics and underlies various concepts, including functions. When working with set theory, remember:

• A set is a well-defined collection of distinct objects, considered as an object in its own right.
• The elements of a set are typically denoted by lowercase letters (e.g., \( a, b, c, d \)).
• Mappings in exercises like the one described often involve sets, where specific elements are paired according to given rules.

In the context of our exercise, understanding set theory helps us comprehend the structure of the mapping \( f: X \rightarrow X \), ensuring each element in the set \( X \) is explicitly accounted for in the function's mapping.

Identifying Mathematical Conditions

Mathematical conditions are specific requirements that must be satisfied for a statement or property to hold true. They play a crucial role in verifying the properties of mathematical objects, such as functions.In the exercise, several conditions need to be fulfilled for the mappings to be valid one-to-one mappings:

• The mapping preserves one-to-one relationships, meaning each input maps to a distinct output.
• Specific stipulations, such as \( f(a) = a \), \( f(b) eq b \), and \( f(d) eq d \), must be satisfied.
• Verifying these conditions involves checking each mapping meticulously to ensure all rules are followed correctly.

By identifying and checking these conditions, the solution ensures that the given mappings meet all necessary criteria, confirming the correctness of the solution.