Discrete Mathematics – Sets, Relations & Functions

What you’ll learn
Introduction
Sets and their Representations
The Empty Set
Finite and Infinite Sets
Equal Sets
Subsets
Power Set
Universal Set
Venn Diagrams
Operations on Sets
Complement of a Set
Practical Problems on Union and Intersection of Two Sets
Types of Relations
Types of Functions
Composition of Functions and Invertible Function
Binary Operations

Requirements
Basic knowledge of mathematics of 9th and 10th std Mathematics

Description
Sets

Sets and their representations

Empty set

Finite and Infinite sets

Equal sets. Subsets

Subsets of a set of real numbers especially intervals (with notations)

Power set

Universal set

Venn diagrams

Union and Intersection of sets

Difference of sets

Complement of a set

Properties of Complement Sets

Practical Problems based on sets

Relations & Functions

Ordered pairs

Cartesian product of sets

Number of elements in the cartesian product of two finite sets

Cartesian product of the sets of real (up to R × R)

Definition of −

Relation

Pictorial diagrams

Domain

Co-domain

Range of a relation

Function as a special kind of relation from one set to another

Pictorial representation of a function, domain, co-domain and range of a function

Real valued functions, domain and range of these functions −

Constant

Identity

Polynomial

Rational

Modulus

Signum

Exponential

Logarithmic

Greatest integer functions (with their graphs)

Sum, difference, product and quotients of functions

SUMMARY

Sets – This chapter deals with some basic definitions and operations involving sets. These are summarised below

1. A set is a well-defined collection of objects. A set which does not contain any element is called empty set.

2. A set which consists of a definite number of elements is called finite set, otherwise, the set is called infinite set.

3. Two sets A and B are said to be equal if they have exactly the same elements.

4. A set A is said to be subset of a set B, if every element of A is also an element of B. Intervals are subsets of R.

5. A power set of a set A is collection of all subsets of A. It is denoted by P(A).

6. The union of two sets A and B is the set of all those elements which are either in A or in B.

7. The intersection of two sets A and B is the set of all elements which are common. The difference of two sets A and B in this order is the set of elements which belong to A but not to B.

8. The complement of a subset A of universal set U is the set of all elements of U which are not the elements of A.

9. For any two sets A and B, (A ∪ B)′ = A′ ∩ B′ and ( A ∩ B )′ = A′ ∪ B′

10. If A and B are finite sets such that A ∩ B = φ, then n (A ∪ B) = n (A) + n (B). If A ∩ B ≠ φ, then n (A ∪ B) = n (A) + n (B) – n (A ∩ B)

Relations & Functions – In this chapter, we studied different types of relations and equivalence relation, composition of functions, invertible functions and binary operations. The main features of this chapter are as follows

1. Empty relation is the relation R in X given by R = φ ⊂ X × X.

2. Universal relation is the relation R in X given by R = X × X.

3. Reflexive relation R in X is a relation with (a, a) ∈ R ∀ a ∈ X.

4. Symmetric relation R in X is a relation satisfying (a, b) ∈ R implies (b, a) ∈ R.

5. Transitive relation R in X is a relation satisfying (a, b) ∈ R and (b, c) ∈ R implies that (a, c) ∈ R.

5. Equivalence relation R in X is a relation which is reflexive, symmetric and transitive.

6. Equivalence class[a] containing a ∈ X for an equivalence relation R in X is the subset of X containing all elements b related to a.

7. A function f : X → Y is one-one (or injective) if f(x1 ) = f(x2 ) ⇒ x1 = x2 ∀ x1 , x2 ∈ X.

8. A function f : X → Y is onto (or surjective) if given any y ∈ Y, ∃ x ∈ X such that f(x) = y.

9. A function f : X → Y is one-one and onto (or bijective), if f is both one-one and onto.

10. The composition of functions f : A → B and g : B → C is the function gof : A → C given by gof(x) = g(f(x)) ∀ x ∈ A.

11. A function f : X → Y is invertible if ∃ g : Y → X such that gof = IX and fog = IY.

12. A function f : X → Y is invertible if and only if f is one-one and onto.

Who this course is for
Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
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