RELATIONS AND FUNCTIONS

Introduction

• G. H. Hardy's Perspective on Mathematical Beauty

  • Emphasizes that there is no permanent place for "ugly mathematics."

  • Mathematical beauty is difficult to define but recognizable, similar to poetry.

• Overview of Relations and Functions

  • Fundamental terms like relations, functions, domain, co-domain, and range were introduced in Class XI.

  • Definition of 'relation' in mathematics: A connection between two objects or quantities.

  • Example of sets: Let A be Class XII students and B be Class XI students.

  • Possible relations include:

    1. {(a, b) ∈ A × B: a is brother of b}

    2. {(a, b) ∈ A × B: a is sister of b}

    3. {(a, b) ∈ A × B: age of a is greater than age of b}

    4. {(a, b) ∈ A × B: total marks obtained by a in final examination is less than those obtained by b}

    5. {(a, b) ∈ A × B: a lives in the same locality as b}

  • An abstract definition of a relation R from A to B as any subset of A × B.

  • Notation: If (a, b) ∈ R, then a is related to b, denoted as a R b.

  • Functions as special types of relations.

Types of Relations

Definition and Examples

• Relations as subsets of A × A.

  • Extreme Cases:

  • Empty set (φ) as an empty relation (R = φ)

  • Universal relation as A × A.

  • Specific Examples:

  • Let A = {1, 2, 3, 4}.

    • Example of empty relation: R = {(a, b): a - b = 10} (no solution exists).

    • Example of universal relation: R' = {(a, b): | a - b | ≥ 0} (true for all pairs).

Types of Relations

1. Empty Relation: R is called an empty relation if R = φ ⊂ A × A.

2. Universal Relation: R is universal if R = A × A.

3. Trivial Relations: Empty and universal relations categorized as such.

Reflexive, Symmetric, and Transitive Relations

• Definition of Reflexive:

  • A relation R in a set A is reflexive if (a, a) ∈ R for every a ∈ A.

• Definition of Symmetric:

  • A relation R is symmetric if (a1, a2) ∈ R implies (a2, a1) ∈ R.

• Definition of Transitive:

  • A relation R is transitive if (a1, a2) ∈ R and (a2, a3) ∈ R implies (a1, a3) ∈ R.

• Equivalence Relation:

  • A relation R is called an equivalence relation if it is reflexive, symmetric, and transitive.

Examples of Relations

Equivalence Relation Example

• Example 1: Let T be a set of triangles.

  • R = {(T1, T2): T1 is congruent to T2} is an equivalence relation.

  • Reflexive: A triangle is congruent to itself.

  • Symmetric: T1 is congruent to T2 implies T2 is congruent to T1.

  • Transitive: T1 congruent to T2 and T2 congruent to T3 implies T1 congruent to T3.

Non-Equivalence Relation Example

• Example 2: Let L be the set of lines in a plane, R = {(L1, L2): L1 is perpendicular to L2}.

  • Not reflexive, since a line cannot be perpendicular to itself.

  • Symmetric: If L1 is perpendicular to L2, then L2 is perpendicular to L1.

  • Not transitive.

  • If L1 ⊥ L2 and L2 ⊥ L3, L1 is not perpendicular to L3; they can be parallel.

Further Examples with Evaluations

• Example 3: R = {(1,1), (2,2), (3,3), (1,2), (2,3)} is reflexive but neither symmetric nor transitive.

• Example 4: R = {(a, b): 2 divides a - b} where R is an equivalence relation in integers, Z.

Equivalence Classes

• Definition: An equivalence class [a] contains elements related to a under equivalence relation R.

• Example: For R defined in integer set Z, subsets E of even integers and O of odd integers are equivalence classes:

  • Each integer in E is related to each other and similarly for O.

  • No even integer is related to any odd integer.

Types of Functions

• Definition and fundamental properties of functions reiterated.

1. One-One (Injective):

   • If f(x1) = f(x2), then x1 = x2.

2. Onto (Surjective):

   • Every element y in co-domain Y is the image of some x in domain X.

3. Bijective:

   • A function is bijective if it is both one-one and onto.

Examples in Functions

• Example functions such as f(x) = 2x shown to be one-one but not onto.

- Distinctions seen in one-one functions from a finite to finite set needing bijection properties.

Composition of Functions

• Definition: The composition of functions f: A → B and g: B → C is given by gof(x) = g(f(x)).

• Demonstrations provided in examples when compositions yield differing results

• Establishment of conditions under which functions are invertible using compositions.

Additional Examples and Exercises

• A multitude of exercises facilitate understanding of reflexive, symmetric, and transitive conditions.

• Exercises encouraging students to examine properties of given relations/functions to ensure comprehension of the subject.

Summary of Chapter

• Key points reiterated regarding relations and functions, emphasizing emptiness, universality, types of relations, equivalence relations, single vs. multiple mappings.

• Importance of proofs and exercises highlighted to bolster understanding of concepts discussed.