CBSE Class XI - Chap2. Relations and Functions

2. 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 (upto R x R). Definition of relation, pictorial diagrams, domain, co-domain and 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 and greatest integer functions, with their graphs. Sum, difference, product and quotients of functions.

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Relations & Functions
https://www.youtube.com/watch?v=OxZ0JL4Bjzk

XI - 2.4 Some results on relations - Video Lectures

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.



Proof of Set operations in Relations -1 / NCERT Std XI Mathematics
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XI - 2.3 Types of relations - Video Lectures

XI -

2.3 Types of relations - Video Lectures

2.3 Types of relations


Void relation
Universal relation
Identity relation
Reflexive relation
Symmetric relation
Transitive relation
Antisymmetric relation
Equivalence relation


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XI - 2.2 Relations - Video Lectures

XI - 2.2 Relations - Video Lectures

2.2 Relation

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.

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XI - 2.1 Cartesian product of sets - Video Lectures

XI - 2.1 Cartesian product of sets - Video Lectures

2.1 Cartesian product of sets

Cartesian product is an operation on sets.


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

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

Theorems

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)
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Ch. 2. Cartesian Product of Sets and Relations - Concepts Review

Contents

2.1 Cartesian product of sets
2.2 Relations
2.3 Types of relations
2.4 Some results on relations
2.5 Composition of relations



2.1 Cartesian product of sets

Cartesian product is an operation on sets.


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

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

Theorems

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)


2.2 Relation

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.


2.3 Types of relations


Void relation
Universal relation
Identity relation
Reflexive relation
Symmetric relation
Transitive relation
Antisymmetric relation
Equivalence relation

Cartesian Products of Sets and Relations - Part 2



2.4 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.


2.5 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^-

Ch.2. Cartesian Product of Sets and Relations - Part 2

Concept Review


2.2 Relation

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.


2.3 Types of relations


Void relation
Universal relation
Identity relation
Reflexive relation
Symmetric relation
Transitive relation
Antisymmetric relation
Equivalence relation


2.4 (2.5 in the book) Composition of Relations

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^-

IIT JEE Study Guide 2. Cartesian Product of Sets and Relations

Objective Mathematics by R D Sharma

Contents

2.1 Cartesian product of sets
2.2 Relations
2.3 Types of relations
2.4 Some results on relations
2.5 Composition of relations

Day 1

2.1 Cartesian product of sets

Day 2

2.2 Relations
2.3 Types of relations

Day 3
2.4 Some results on relations
2.5 Composition of relations

Day 4
Objective Exercises 1 to 37

Day 5
Fill in the blanks 1 to 8
True/False 1 to 13


Day 6
Practice exercises 1 to 27

Revision days


Day 7

Day 8

Day 9

Day 10