Relations and Functions

1.1 Introduction

This section revisits the foundational concepts of relations and functions introduced in Class XI, emphasizing the notions of domain, co-domain, and range. The term 'relation' in mathematics is inspired by its English meaning, where two objects or quantities are related if there exists a recognizable connection between them. For example, consider two sets: A, the set of Class XII students in a school, and B, the set of Class XI students of the same school. Several relations from A to B can be defined, such as 'a is brother of b', 'a is sister of b', 'age of a is greater than age of b', 'total marks obtained by a is less than that of b', or 'a lives in the same locality as b'. Mathematically, a relation R from A to B is defined as any subset of the Cartesian product A × B. If (a, b) ∈ R, we say 'a is related to b' under relation R and denote it as aRb. Unlike everyday language, in mathematics, the existence of a relation does not necessarily require a recognizable connection. Functions are special types of relations where each element of the domain is related to exactly one element of the co-domain. This chapter aims to explore various types of relations and functions, including composition and invertibility, as well as binary operations.

• Relation is a subset of Cartesian product A × B.
• If (a, b) ∈ R, then a is related to b under relation R.
• Functions are special relations with unique images for each domain element.
• Examples of relations include familial ties, age comparisons, and locality.
• Domain, co-domain, and range are key concepts linked to functions.
• The chapter will study types of relations, functions, composition, and invertibility.
• 📌 Relation: A subset of Cartesian product A × B linking elements of A to B.
• 📌 Function: A special relation where each element of domain maps to exactly one element of co-domain.
• 📌 Domain: Set of all inputs for a function.

1.2 Types of Relations

Relations within a set A are subsets of A × A. Two extreme cases are the empty relation and the universal relation. The empty relation has no related pairs, i.e., R = ∅, while the universal relation relates every element to every other element, i.e., R = A × A. For example, in the set A = {1, 2, 3, 4}, the relation R = {(a, b) : a - b = 10} is empty since no such pair exists, whereas R' = {(a, b) : |a - b| ≥ 0} is universal as all pairs satisfy this. These trivial relations help define more complex properties of relations: reflexivity, symmetry, and transitivity. A relation R is reflexive if every element is related to itself, symmetric if whenever a is related to b, b is related to a, and transitive if whenever a is related to b and b to c, then a is related to c. An equivalence relation is one that is reflexive, symmetric, and transitive. Examples include congruence of triangles (equivalence relation), perpendicularity of lines (symmetric but neither reflexive nor transitive), and divisibility relations on integers. Equivalence relations partition the set into equivalence classes, subsets where elements are related to each other but not to elements outside the subset. For instance, integers can be partitioned into even and odd classes under the relation 'difference divisible by 2'. This section also introduces notation aRb to denote that a is related to b under relation R.

• Empty relation R = ∅ relates no elements.
• Universal relation R = A × A relates every element to every element.
• Reflexive: (a, a) ∈ R for all a ∈ A.
• Symmetric: (a, b) ∈ R implies (b, a) ∈ R.
• Transitive: (a, b) ∈ R and (b, c) ∈ R imply (a, c) ∈ R.
• Equivalence relation is reflexive, symmetric, and transitive.
• 📌 Empty Relation: Relation with no related pairs.
• 📌 Universal Relation: Relation relating every element to every element.
• 📌 Reflexive Relation: Relation where every element relates to itself.