Class 12 - Chapter 1

Relations

Empty Relation (Trivial Relation): A relation R in a set A where no element of A is related to any element of A, i.e., $R = \phi \subset A \times A$ .
Universal Relation (Trivial Relation): A relation R in a set A where each element of A is related to every element of A, i.e., $R = A \times A$ .
Types of Relations:
- Reflexive: A relation R in a set A is reflexive if $(a, a) \in R$ , for every $a \in A$ .
- Symmetric: A relation R in a set A is symmetric if $(a_1, a_2) \in R$ implies that $(a_2, a_1) \in R$ , for all $a_1, a_2 \in A$ .
- Transitive: A relation R in a set A is transitive if $(a_1, a_2) \in R$ and $(a_2, a_3) \in R$ implies that $(a_1, a_3) \in R$ for all $a_1, a_2, a_3 \in A$ .
Equivalence Relation: A relation R in a set A is an equivalence relation if R is reflexive, symmetric, and transitive.
Equivalence Class [a]: For an equivalence relation R in X, the equivalence class $[a]$ containing $a \in X$ is the subset of X containing all elements b related to a. An equivalence relation partitions the set X into mutually disjoint equivalence classes.

Functions

Types of Functions:
- One-one (or Injective): A function $f:X \rightarrow Y$ is one-one if the images of distinct elements of X under $f$ are distinct, i.e., for every $x_1, x_2 \in X$ , $f(x_1) = f(x_2)$ implies $x_1 = x_2$ . Otherwise, it is called many-one.
- Onto (or Surjective): A function $f:X \rightarrow Y$ is onto if every element of Y is the image of some element of X under $f$ , i.e., for every $y \in Y$ there exists an element $x$ in X such that $f(x) = y$ . This is equivalent to the Range of $f$ being equal to Y.
- One-one and onto (or Bijective): A function $f:X \rightarrow Y$ is bijective if it is both one-one and onto.

Composition of Functions and Invertible Function

Composition of Functions: For two functions $f:A \rightarrow B$ and $g:B \rightarrow C$ , the composition of $f$ and $g$ , denoted by $gof$ , is defined as the function $gof: A \rightarrow C$ given by $gof(x) = g(f(x))$ , $\forall x \in A$ .
Invertible Function: A function $f:X \rightarrow Y$ is invertible if there exists a function $g:Y \rightarrow X$ such that $gof = I_X$ and $fog = I_Y$ . The function $g$ is called the inverse of $f$ and is denoted by $f^{-1}$ . A function is invertible if and only if it is one-one and onto.

Characteristic Property of a Finite Set

For a finite set X, a function $f:X \rightarrow X$ is one-one if and only if it is onto. This is not true for an infinite set.

Category 1: Relations

This category focuses on determining the properties of a given relation $R$ defined on a set $A$ , or proving that a relation possesses certain properties.

Sub-Category	Focus	Pattern of Questions (from Exercise 1.1)
1A: Basic Properties of Relations	Testing if a relation is Reflexive, Symmetric, and/or Transitive	Questions 1, 2, 3, 4, 5, 6, 10: Given a set $A$ and a rule for $R$ , determine which of the three properties ( $R$ , $S$ , $T$ ) it satisfies. Example: $R = {(x,y): 3x - y = 0}$ on $A = {1, 2, ..., 14}$ .
1B: Equivalence Relations and Classes	Proving a relation is an Equivalence Relation (must be R, S, and T) and finding the resulting Equivalence Classes or sets of related elements.	Questions 7, 8, 9, 11, 12, 13, 14: Prove $R$ is an equivalence relation. Then, identify all elements related to a specific element (e.g., all elements related to 1, or the circle passing through $P$ ). Example: $R = {(x,y):
1C: Multiple Choice/Specific Examples	Selecting the correct classification of a given relation or finding a specific element that satisfies the relation's rule.	Questions 15, 16: Choose the correct option that describes the properties of a given relation $R$ , or find the correct ordered pair that belongs to $R$ .
1D: Miscellaneous/Theoretic	Proving properties related to multiple relations (e.g., intersection) or counting the number of relations with specific properties.	Miscellaneous Examples 18, 23, 24: Proving $R_1 \cap R_2$ is an equivalence relation, or counting the number of relations/equivalence relations satisfying certain criteria.

Category 2: Functions

This category focuses on classifying a given function $f$ as one-one (injective), onto (surjective), or bijective, and calculating the composition or inverse of functions.

Sub-Category	Focus	Pattern of Questions (from Exercise 1.2 & 1.3)
2A: Types of Functions (Injectivity and Surjectivity)	Determining if a function $f: X \to Y$ is One-one (Injective) or Onto (Surjective) by direct proof or counterexample, and classifying it as Bijective (if both).	Questions 1, 2, 6, 7, 9, 10, 11, 12: Check for injectivity and surjectivity for functions defined on $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{R}$ , or specific finite sets. Example: $f(x)=x^2$ or $f(x)=3-4x$ on $\mathbb{R}$ .
2B: Special Functions	Examining the injectivity and surjectivity of well-known special functions.	Questions 3, 4, 5: Prove the properties for the Greatest Integer Function $f(x) = [x]$ , the Modulus Function $f(x) =
2C: Function Composition	Calculating the Composition of Functions ( $f \circ g$ and $g \circ f$ ).	Example 16: Given $f(x)$ and $g(x)$ , find $g(f(x))$ and $f(g(x))$ .
2D: Invertible Functions	Determining if a function is Invertible (must be one-one and onto) and finding the Inverse Function $f^{-1}$ .	Example 17: Given $f(x)$ , show it is invertible and find $g(y) = f^{-1}(y)$ .

Pattern of Questions Asked

The questions in the chapter follow a systematic progression, moving from definitional checks to comprehensive proofs and application of theorems:

Definitional Checks (Basic Level): The majority of questions (especially in Exercise 1.1) require the student to check if a relation satisfies the three fundamental definitions: Reflexive, Symmetric, and Transitive. Similarly, for functions (Exercise 1.2), the pattern is to check Injectivity and Surjectivity.
Equivalence Proofs (Medium Level): A significant number of problems require proving that a relation is an Equivalence Relation, which mandates proving all three properties (R, S, T) simultaneously (e.g., questions involving congruence, similarity, or same number of pages/sides).
Specific Function Analysis: Questions often test the application of function definitions on different domains ( $\mathbb{N}, \mathbb{Z}, \mathbb{R}$ ) and for specific types of functions (e.g., $x^2, x^3$ , Signum, Greatest Integer, Modulus). This highlights that the domain/co-domain significantly impacts the function's type.
Theoretic Application (Advanced Level): The final set of problems (Examples 18-26, Miscellaneous Exercise) involves abstract proofs (e.g., properties of intersection of equivalence relations), counter-examples for composite/combined functions, or characteristic properties of finite sets (where one-one implies onto).

The questions in the chapter "RELATIONS AND FUNCTIONS" can be categorized based on the concepts they test, separating them by exercise (Exercise 1.1, Exercise 1.2, and Miscellaneous Examples/Exercise).I. Relations (Exercise 1.1)

This exercise focuses entirely on determining the properties of a relation $R$ defined on a given set, or proving that $R$ is an equivalence relation.

Category	Questions (by Number)	Focus / Solution Method
1. Check Reflexive, Symmetric, Transitive (R, S, T)	1, 2, 3, 4, 5, 6, 10	The core of the exercise. These questions require checking the three basic definitions using the relation rule (e.g., $3x-y=0$ , $a \le b^2$ , $x-y$ is an integer). Question 10 requires generating examples for specific combinations of R, S, and T properties.
2. Prove Equivalence Relation	7, 8, 9, 11, 12, 13, 14	These questions require proving that the relation is Reflexive, Symmetric, AND Transitive.
3. Equivalence Class / Related Elements	8, 9, 11, 12, 13, 14	These questions follow an equivalence proof and require identifying the subset of elements related to a specific element (the equivalence class). Examples: elements related to 1 (Q9), or the circle passing through point P (Q11).
4. Multiple Choice Questions (MCQ)	15, 16	These are quick checks on the properties (R, S, T) for a given relation, or selecting a pair that satisfies the relation's rule.

II. Functions (Exercise 1.2)

This exercise focuses on classifying functions as one-one (injective), onto (surjective), or bijective.

Category	Questions (by Number)	Focus / Solution Method
1. Check Injectivity (One-one) and Surjectivity (Onto)	1, 2, 6, 7, 9, 10	These questions require proving or disproving one-one (by showing $f(x_1) = f(x_2) \Rightarrow x_1 = x_2$ ) and onto (by showing that for every $y$ in the co-domain, an $x$ exists in the domain such that $f(x)=y$ ) for various function types and domains ( $\mathbb{N}$ , $\mathbb{Z}$ , $\mathbb{R}$ , $\mathbb{R}_{*}$ ).
2. Special Function Analysis	3, 4, 5	Focuses on proving the properties (or lack thereof) for standard special functions: Greatest Integer Function ( $[x]$ ), Modulus Function ($
3. Proving Bijective	7 (i), 8	Requires proving the function is both one-one and onto. Question 8 deals with a function between product sets $A \times B \to B \times A$ .
4. Multiple Choice Questions (MCQ)	11, 12	Quick classification of functions like $f(x) = x^4$ and $f(x)=3x$ on $\mathbb{R}$ based on their type (one-one, onto, neither, or both).

III. Composition, Inverse, and Miscellaneous (Section 1.4 & Miscellaneous)

This section covers the composition of functions, finding the inverse, and theoretical problems involving equivalence relations and function properties on finite sets.

Category	Questions (by Number)	Focus / Solution Method
1. Composition of Functions ( $gof$ and $fog$ )	Example 15, Example 16	Calculating the resulting function when two functions $f$ and $g$ are combined. Example 16 specifically shows that $gof \ne fog$ .
2. Invertible Functions	Example 17	Proving a function is invertible (by establishing it is one-one and onto) and finding the Inverse Function $f^{-1}$ .
3. Theoretical Proofs on Relations	Examples 18, 19, 20, 21, Miscell. 3	Proofs that involve advanced properties of relations, such as: - Intersection of two equivalence relations (Ex. 18). - Equivalence relation on ordered pairs (Ex. 19). - Equivalence relation induced by a function (Ex. 21). - Checking if subset relation ( $A \subset B$ ) on $P(X)$ is an equivalence relation (Miscell. 3).
4. Theoretical Proofs/Counting on Functions	Examples 22, 23, 24, 25, 26, Miscell. 4, 5	Problems involving counting functions or exploring function properties on finite sets: - Counting the number of one-one functions (permutations) (Ex. 22). - Counting relations with specific properties (Ex. 23, 24, Miscell. 6, 7). - Showing properties do not hold for composite/combined functions (Ex. 25, 26).
5. Domain/Range Bijectivity	Miscell. 1, 2	Proving injectivity/bijectivity for functions with specific restricted domains and co-domains.

Class 12 - Chapter 1

Category 1: Relations

Category 2: Functions

Pattern of Questions Asked

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