Real Functions - Chapter Revision Points

Sections in the Chapter

1.Introduction

2 Description of real functions

3 Intervals (Closed and open)

4 Domains and ranges of real functions

5 Real functions - Examples

6 Operations on real functions

7 Even and odd functions

8 Extension of a function

9 Periodic function


1 Description of real functions

2 Intervals (Closed and open)

3 Domains and ranges of real functions

4 Real functions - Examples

5 Operations on real functions

6 Even and odd functions

7 Extension of a function

8 Periodic function




If the domain and co-domain of a function are subsets of R (set of all real numbers), it is called a real valued function or in short a real function.


Updated 17 Jan 2016, 2 Dec 2008

Limits - Chapter Revision Points

Sections in the Chapter

20.1 Informal approach to limit
20.2 Formal approach to limit
20.3 Evaluation of left hand and right hand limits
20.4 Difference between the value of a function at a point and the limit at a point
20.5 The algebra of limits
20.6 Evaluation of limits




We can approach a given number ‘a’ on the real line from its left hand side by increasing numbers which are less than ‘a’. It means starting from a- δ and increasing to reach a.

We can also approach a given number ‘a’ on the real line from its right hand side by decreasing numbers which are greater than ‘a’. It means starting from a+δ and decreasing to reach a.

Hence there are two types of limits – left hand limit and right hand limit.

For some functions both these limits are equal at a point and for some functions they are not equal.

If both are equal we say lim (x→a) f(x) exists. Otherwise it does not exist.


Updated 17 Jan 2016, 2 Dec 2008

Set: Explanation

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z⁺: positive integers
Q: rational numbers
Q⁺: positive rational numbers
R: real numbers
R⁺: positive real numbers
C: complex numbers

Description of a set

Sets can be described by roster method or set-builder method.

Roster method:
In this method, the set is described by listing all the elements within braces { }, separated by commas.

Example: {2,4,6,8,10}
It is a set having 5 elements.

Set-builder method:
In this method, a set is described by a property of x where x represents the elements. If the property of x is represented by P(x), the set description is given by
{x : P(x) is satisfied} or {x| P(x) is satisfied}

Example: {x| x is an even number less than or equal to 10}
This description will give {2,4,6,8,10} in roster form.

Types of sets

Empty set (ф)
A set is said t be empty or null or void set if it has no element and it is denoted by ф.

Singleton set
A set consisting of single element.

Finite set
A set is called a finite set if it is either void set or its elements can be listed (counted or labeled) by natural numbers 1,2,3 … and the counting of number of elements stops at a certain natural number of say (n).

The number of elements in a finite set (n) is called the cardinal number or order of a finite set A and is denoted by n(A).

Infinite set
A set who elements cannot be listed by the natural numbers however large the number may be is called an infinite set.

Equivalent set
Two finite sets are equivalent if their cardinal numbers or number of elements are same.

Equal set
Two sets A and B are equal if every element in A is a member of B and every element of B is a member of A.

Subset
When A and B are two sets, if every element of A is an element of B, then A is called a subset of B.

Universal set (U)
In discussions of sets, the superset that contains all other sets in discussion is called the universal set.

Power set
When A is a set, the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

Power set is a set of subsets or elements of a power set are subsets of a set.
P(A) = {S: S is a subset of A}

If A is a finite set having n elements, the P(A) has 2ⁿ elements.


Complement of a set
If U is a universal set, the complement of a set A with respect to U is denoted as A’ or A^c or U – A . It is a set of those elements of U which are not in A.

A’ = {x| x є U, and x is does not belong to A}

Theorems on subsets

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ

Operations on sets

Union of sets
The union of sets A and B is th set of all those elements which belong either to A or to B or to both A and B.

The symbol used to denote union of sets and A and B is A U B.

x Є (A U B) implies x Є A or x Є B
x does not belong to A U B implies x does not belong to A and also x does not belong to B.

Intersection of sets
The intersection of sets A and B is the set of all those elements that belong to both A and B.

The intersection of sets A and B is denoted by A ∩ B.

x Є (A ∩ B) implies x Є A and also x Є B.


Difference of sets
The difference of sets A and B, written as A-B is the set of all those elements of A which do not belong B.

It means x Є (A - B) implies x Є A or x does not belong to B.

Symmetric difference of sets
The symmetric difference of sets A and B is the set (A-B)U((B-A) and is denoted by AΔB.

AΔB = (A-B)U((B-A) = {x: x does not belong to A ∩ B).

Types of sets based on operations

Disjoint sets

Complement of a set
When U is the universal set and A is a subset of U, the complement of A with respect to U is denoted by A’ or A₀ or U-A and it is defined as the set of all those elements of U which are not in A.

A’ = {x: x does not belong to A but x ЄU}
x ЄA’ implies x does not belong to A.

Laws of algebra of sets

1. Idempotent laws

(i) A U A = A
(ii) A ∩ A = A

2. Identity laws

(i) A U ф = A
(ii) A ∩ U = A

3. Commutative law
(i) A U B = B U A
(ii) A ∩ B = B ∩ A

4. Associative laws

(i) (A U B) U C = A U (B U C)
(ii) (A ∩ B) ∩ C = A ∩ (B ∩ C)

5. Distributive laws

(i) A U (B ∩ C) = (A U B) ∩ (A U C)
(ii) A ∩ (B U C) = (A ∩ B) U (A ∩ C)

6. De-morgan’s laws

(i) (A U B)’ = A’ ∩ B’
(ii) (A ∩ B)’ = A’ U B’

Some more deductions/theorems/ related to operations on sets

If A and B are two sets

(i) A – B = A ∩ B’
(ii) B – A = B ∩ A’
(iii) A – B = A <=> A ∩ B = ф
(iv) (A – B) U B = A U B
(v) (A-B) ∩ B = ф
(vi) A is a sub set of B <=> B’ is a subset of A’
(vii) (A-B) U (B-A) = (A U B) – (A ∩ B)

If A, B and C are three sets, then

(i) A – (B ∩ C) = (A-B) U (A-C)
(ii) A – (B U C) = (A-B) ∩ (A-C)
(iii) A ∩ (B-C) = (A ∩ B) - (A ∩ C)
(iv) A ∩ (B Δ C) = (A∩B) Δ (A∩C)

Number of elements in sets n(A) and Some Results on Them

Note union operation and universal set have the same symbol in these pages. Hence identify appropriately.

n(A) denotes the number of elements in the set A. Similarly n(B) and n(C).

If A,B and C are finite sets. U is the finite universal set, then

(i) n(A U B) = n(A) +n(B) – n(A∩B)

(ii) n(A U B) = n(A) +n(B) <=> A, B are disjoint non-void sets.

(iii) n(A-B) = n(A) –n(A∩B)

(iv) n(A ΔB) = Number of elements which belong to exactly one of A or B
= n((A-B) U (B-A))

(v) n(A U B U C) = n(A) + n(B) + n(C) – n(A∩B) – n(B∩C) – n(A∩C)+n(A∩B∩C)

(vi) No. Of elements in exactly two of the sets A,B,C
= n(A∩B) + n(B∩C)+n(C∩A)-3n(A∩B∩C)

(vii) No. Of elements in exactly one of the sets A,B,C
= n(A) +n(B)+n(C)-2n(A∩B)-2n(B∩C)-2n(A∩C)+3n(A∩B∩C)

(viii) n(A’ U B’) = n((A∩B)’) = n(U) – n(A∩B)

(ix) n(A’∩B’) = n((AUB)’) = n(U)-n(A∩B)

Ordered Pair

An Ordered pair consists of two objects or elements in a given fixed order.

For example when A and B are any two sets, a pair (a,b) where a ЄA and b ЄB is an ordered pair. The fixed order comes from the two sets A and B and the first element is from A and the second element is from B.

Equality of ordered pairs: Two ordered pairs (a1,b1) and (a2,b2) are equal to if a1 = a2 and b1 = b2.

Cartesian product of sets

Cartesian product is an operation on sets.

Let A and B be any two non empty sets. The set of all ordered pairs (a,b) such that aЄA and bЄB is called the Cartesian product of the sets A and B and is denoted by A×B

A×B represents Cartesian product of sets and it is a set of ordered pairs.

Theorems on Cartesian product of sets

Theorem 1; For any three sets

(i) A×(B U C) = (A×B) U (A×C)
(ii) A×(B∩C) = (A×B) ∩(A×C)

Theorem 2: For any three sets

A×(B – C) = (A×B) – (A×C)

Theorem 3: If and A and B are any two non-empty sets, then

A×B = B×A  A = B

Theorem 4: If A is a subset of B, A×A is a sub set of (A×B) ∩(B×A)

Theorem 5
If A is a subset of B, (A×C) is a subset of (B×C) for any set C.

Theorem 6
If A is a subset of B and C is a subset of D, (A×C) is a subset of (B×D)

Theorem 7
For any sets A,B,C , D,

(A×B) ∩(C×D) = (A∩C) ×(B∩D)

Theorem 8

For any three sets A,B,C

i. (A×(B’ U C’) = (A×B) ∩(A×C)
ii. (A×(B’ ∩ C’) = (A×B) U(A×C)

Theorem 9
When A and B are two non-empty sets having n elements in common, (A×B) and (B×A) will have n² elements in common.

Relation - Definition

Let A and B be two sets. Then a relation R from A to B is a subset of A×B.

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

Total number of relations: If A and B are two non empty sets with m and n elements respectively, A×B consists of mn ordered pairs.
Since each subset defines a relation from A to B, so total number of relations from A to B is 2^mn.

Domain and Range of Relation

R is a relation means that it is a set of ordered paris.


Domain of a relation

When R is a relation from a set A to set B, the set of all first components of the ordered pairs belonging to R is called the domain of R.

Range of a relation

When R is a relation from a set A to set B, the set of all second components of the ordered pairs belonging to R is called the range of R.

In a relation from set A to set B, domain of the relation will be a subset of A and range of the relation will be a subset of B.

Relation on Set A and Inverse Relation

Relation on Set A
A relation from A to A i.e., a subset of A×A, is called a relation on set A.

Inverse relation
When a relation R is from set A to set B, a relation from set B to set A denoted by R^-1 is the inverse of R. That is if R is a set of (a,b), then R^-1 is a set of (b,a).

In the case Domain of relation R = Range of relation R^-1

Range of relation R = Domain of relation R^-1

Types of relations

Void or empty relation
When A is a set, ф is a subset of A×A and so it is a relation on A. This relation is called the void or empty relation on A.

Universal relation
When A is a set, A×A is a subset of A×A and so it is a relation on A. This relation is called the universal relation.

Identity relation
When A is a set, the relation I_A = {(a,a):a ЄA} on A is called the identity relation on A.

Reflexive relation
A relation on a set A is said to be reflexive if every element of A is related to itself.

Think. What is the difference between identity relation and reflexive relation?

Symmetric relation
A relation R on a set A is said to be a symmetric relation iff

(a,b) Є R implies (b,a) ЄR for all a,b Є A.

i.e. aRb implies bRa for a,b Є.

Transitive relation
A relation R on a set A is said to be a transitive relation iff
(a,b) ЄR and (b,c) ЄR implies (a,c) ЄR for a,b,c ЄA.

Antisymmetric relation
A relation R on set A is said t be an antisymmetric relation iff
(a,b) Є R and(b,a) Є R implies a =b for all a,b ЄA.

Equivalence relation
A relation R on set A is said to be an equivalence relation on A iff
i. it is reflexive.
ii. it is symmetric
iii. it is transitive.

Some more properties and results on relations

1. If R and S are two equivalence relations on a set A, then R∩S is also an equivalence relation on A.
2. The union of two equivalence relations on a set is not necessarily an equivalence relation on the set.
3. If R is an equivalence relation on a set A, the R^-1 is also an equivalence relation on A.

Composition of Relations

When r and S are two relations from set A to B and B to C respectively, we can define a relation SoR from A to C such that

(a.c) Є SoR imples for all b Є B subject to the relations (a,b) ЄR and (b.c) ЄS.

SoR is called the composition of R and S.

Properties of SoR

In general RoS is not equal to SoR.

(SoR) ^- = R^-oS^-