Discrete Mathematics - Sets, Relations & Functions

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.

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