Discrete Mathematics – Sets, Relations & Functions
This course offers a comprehensive study of sets, relations, and functions, along with their representations and operations. It covers topics such as empty sets, finite and infinite sets, equal sets, subsets, power sets, Venn diagrams, and practical problems based on sets. Additionally, it delves into types of relations, types of functions, composition of functions, invertible functions, and binary operations. This course is designed for adults preparing for engineering entrance exams, board exams, and competitive exams worldwide. Suitable for learners from over 160 countries, including Europe, America, Middle East, Asia, Africa, and APAC.
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
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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.
“This course is for Adults”
Who this course is for:
- Complete Mathematics for Engineering Entrance Exam Preparation. ( IIT-JEE Main | Advanced | BITSAT | SAT | etc.)
- Those preparing for board and competitive exams State Board, CBSE, ICSE , IGCSE, MHT-CET & NEET
- Courses are suitable for 160 countries from Europe, America, Middle East, Asia, Africa and APAC. Notably England, Germany, France, Sweden, Ireland, Scotland, USA, Canada, UAE, Saudi, Qatar, Kuwait, Malaysia, Indonesia, Myanmar, Newzealand, Australia, South Africa, South Korea, Nigeria, Nepal, Sri Lanka, etc