Topic 1: Sets, Relations, and Functions

This chapter is fundamental and lays the base for many other mathematical concepts. Here’s a detailed explanation:

A. Sets

1. Definition:

A set is a well-defined collection of distinct objects, considered as an object in its own right.

2. Representation of Sets:

• Roster Form (Tabular Form): Listing all elements, e.g., A = {1, 2, 3, 4}

• Set-builder Form: Describes properties, e.g., A = {x | x is a natural number less than 5}

3. Types of Sets:

• Empty Set (∅): No elements, e.g., set of prime numbers divisible by 2 and 3.

• Finite and Infinite Sets

• Equal Sets

• Subsets: A ⊆ B means every element of A is in B.

• Power Set (P(A)): Set of all subsets of A.

• Universal Set (U): Contains all objects under consideration.

• Disjoint Sets: A ∩ B = ∅

4. Venn Diagrams:

Used to represent sets graphically.

5. Operations on Sets:

• Union (A ∪ B): All elements in A or B or both.

• Intersection (A ∩ B): Elements common to A and B.

• Difference (A – B): Elements in A but not in B.

• Complement (A’): Elements in U but not in A.

B. Relations

1. Cartesian Product:

If A = {1, 2}, B = {a, b}, then
A × B = {(1, a), (1, b), (2, a), (2, b)}

2. Definition of Relation:

A relation R from A to B is a subset of A × B.

3. Types of Relations:

• Reflexive: (a, a) ∈ R for all a ∈ A

• Symmetric: If (a, b) ∈ R, then (b, a) ∈ R

• Transitive: If (a, b) ∈ R and (b, c) ∈ R, then (a, c) ∈ R

• Equivalence Relation: A relation that is reflexive, symmetric, and transitive.

C. Functions

1. Function Definition:

A function is a special relation where each element of set A has exactly one image in set B.

2. Types of Functions:

• One-One (Injective): Different inputs give different outputs.

• Onto (Surjective): Every element of the codomain has a pre-image.

• Bijective: Both one-one and onto.

• Constant Function: f(x) = c

• Identity Function: f(x) = x

3. Domain, Co-domain, and Range:

• Domain: Set of all input values

• Co-domain: Set containing all possible output values

• Range: Actual output values from the function

Important Notes:

• Number of subsets of a set with n elements = 2ⁿ

• De Morgan’s Laws:

  • (A ∪ B)’ = A’ ∩ B’

  • (A ∩ B)’ = A’ ∪ B’