Mathematics Crash Course JEE Main>Set Theory and Relations>Level 2>Q 1
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1. Basics of Sets:
(i) A set is a collection of well-defined objects which are distinct from each other.
A set is usually denoted by capital letters A,B,C…. etc. and the elements of the set by a,b,c… etc.
If a is an element of a set A, then we write a∈A and say a belongs to A.
If a does not belong to A, then we write a∉A.
(ii) Methods to represent a set:
(a) Roster method: In this method a set is described by listing elements, separated by commas and enclosed in curly brackets.
(b) Set builder form: In this case we write down a property or rule which gives all the elements of the set.
(iii) Types of sets:
(a) Null set or empty set: A set having no element in it is called an empty set or a null set. It is denoted by ϕ or .
(b) Non-empty set: A set consisting of at least one element is called a non-empty set.
(c) Singleton set: A set consisting of a single element is called a singleton set.
(d) Finite set: A set which has only finite number of elements is called a finite set.
Order of a finite set - The number of elements in a finite set is called the order of the set A and is denoted by nA. It is also called cardinal number of the set.
(e) Infinite set: A set which has an infinite number of elements is called an infinite set.
(f) Equal sets: Two sets A and B are said to be equal if every element of A is a member of B and every element of B is a member of A. If sets A and B are equal, then we write A=B and if A and B are not equal, then A≠B.
(g) Equivalent sets: Two finite sets A and B are equivalent if their number of elements are same i.e., nA=nB.
2. Subsets and Power Set:
(i) Subsets: Let A and B be two sets if every element of A is an element of B, then A is called a subset of B and we write A⊆B.
(ii) Proper subset: If A is a subset of B and A≠B, then A is a proper subset of B and we write A⊂B. The total number of subsets of a finite set containing n elements is 2n.
(iii) Universal set: A set consisting of all possible elements which occur in the discussion is called a universal set
and is denoted by U.
(iv) Power set: Let A be any set, then the set of all subsets of A is called power set of A and is denoted by PA.
3. Operations on Set:
(i) Union of sets:
A union of two or more sets has all their elements. It is denoted by ∪.
(ii) Intersection of sets:
An intersection of two or more sets has the elements which are common to all the sets. It is denoted by ∩.
(iii) Difference of sets:
The difference of the sets A and B in the same order is the set of elements which belong to A but does not belong to B. It is denoted by A-B.
(iv) Symmetric difference of sets:
The symmetric difference of the sets A and B is given by A-B∪B-A. It is denoted by A∆B.
4. Algebra of sets:
(i) Commutative law:
For any two finite sets A and B;
(a) A∪B=B∪A
(b) A∩B=B∩A
(ii) Associative law:
For any three finite sets A, B and C;
(a) A∪B∪C=A∪B∪C
(b) A∩B∩C=A∩B∩C
Thus, union and intersection are associative.
(iii) Idempotent law:
For any finite set A;
(a) A∪A=A
(b) A∩A=A
(iv) Distributive law:
For any three finite sets A, B and C;
(a) A∪B∩C=A∪B∩A∪C
(b) A∩B∪C=A∩B∪A∩C
Thus, union and intersection are distributive over intersection and union respectively.
(v) De Morgan’s laws:
(a) A∪B'=A'∩B'
(b) A∩B'=A'∪B'
(vi) More laws of algebra of sets:
(a) For any two finite sets A and B;
A-B=A∩B'
B-A=B∩A'
A-B=A⇔A∩B=ϕ
A-B∪B=A∪B
A-B∩B=ϕ
A⊆B⇔B'⊆A'
A-B∪B-A=A∪B-A∩B
(b) For any three finite sets A, B and C;
A-B∩C=A-B∪A-C
A-B∪C=A-B∩A-C
A∩B-C=A∩B-A∩C
A∩B△C=A∩B△A∩C
5. Some important results on number of elements in sets:
If A, B and C are finite sets, and U be the finite universal set, then
(i) nA∪B=nA+nB-nA∩B
(ii) nA∪B=nA+nB⇔A,B are disjoint non-void sets.
(iii) nA-B=nA-nA∩B i.e. nA-B+nA∩B=nA
(iv) nA△B= Number of elements which belong to exactly one of A or B=nA-B∪B-A
=nA-B+nB-A [ ∵A-B and B-A are disjoint]
=nA-nA∩B+nB-nA∩B
=nA+nB-2nA∩B
(v) nA∪B∪C=nA+nB+nC-nA∩B-nB∩C-nA∩C+nA∩B∩C
(vi) Number of elements in exactly two of the sets A, B and C
=nA∩B+nB∩C+nC∩A-3nA∩B∩C
(vii) Number of elements in exactly one of the sets A, B and C
=nA+nB+nC-2nA∩B-2nB∩C-2nA∩C+3nA∩B∩C
(viii) nA'∪B'=nA∩B'=nU-nA∩B
(ix) nA'∩B'=n(A∪B')=nU-nA∪B
6. Intervals as Subsets of R:
Assume a,b∈R and a<b.
So, a,b=x:a<x<b and a,b=x:a≤x≤b
7. Cartesian Product of Sets:
For two non-empty sets A and B, the Cartesian product is denoted by A×B and it is the set of all possible ordered pairs (a,b), where a∈A and b∈B i.e., A×B={(a,b)∣a∈A and b∈B}.
(i) Properties of Cartesian product:
(a) The Cartesian product is non-commutative: A×B≠B×A.
(b) A×B=B×A, if and only if A=B.
(c) A×B=ϕ, if either A=ϕ or B=ϕ.
(d) The Cartesian product is non-associative: A×B×C≠A×B×C.
(e) Distributive property over set intersection: A×B∩C=A×B∩A×C.
(f) Distributive property over set union: A×B∪C=A×B∪A×C.
(g) Distributive property over set difference: A×B-C=A×B-(A×C)
(h) If A⊆B, then A×C⊆B×C for any set C.
8. Relations and its Types:
(i) Relations: Let A and B be two sets. Then a relation R from A to B is a subset of A×B, thus R is a relation from A to B⇔R⊆A×B.
(ii) Number of Relations: Let A and B be two non-empty finite sets consisting of m and n elements, respectively. Then, A×B consists of mn ordered pairs. So, total number of subsets of A×B is 2mn.
(iii) Domain and Range of a relation: Let R be a relation from a set A to a set B. Then, the set of all first components of the ordered pairs belonging to R is called the domain of R, while the set of all second components or coordinates of the ordered pairs in R is called the range of R.
Thus, Domain R=a:a, b∈R and Range R=b:a, b∈R.
The domain of a relation from A to B is a subset of A and its range is a subset of B.
9. Types of Relations:
(i) Void Relation: Let A be a set. Then, ϕ⊆A×A and so it is a relation on A and this relation is called the void or empty relation in A.
(ii) Universal Relation: Let A be a set. Then, A×A⊆A×A and so it is a relation on A and this relation is called the universal relation on A.
(iii) Identity Relation: Let A be a set. Then, the relation IA=a,a:a∈A on A is called the identity relation on A i.e., a relation IA on A is called the identity relation if every element of A is related to itself only.
(iv) Reflexive Relation: A relation R on a set A is said to be reflexive if every element of A is related to itself.
(v) Symmetric Relation: A relation R on a set A is said to be a symmetric relation iff a,b∈R ⇒b,a∈R for all a,b∈A i.e., aRb ⇒bRa for all a,b∈A.
(vi) Transitive Relation: A relation R on a set A is said to be a transitive relation iff a,b∈R and b,c∈R⇒a,c∈R for all a,b,c∈A i.e., aRb and bRc⇒aRc for all a,b,c∈A.
(vii) Antisymmetric Relation: Let A be any set. A relation R on set A is said to be an antisymmetric relation iff a,b ∈ R and b,a ∈R ⇒ a=b for all a,b, c ∈ A.
(viii) Equivalence Relation: A relation R on a set A is said to be an equivalence relation on A iff
(a) It is reflexive i.e., a,a∈R for all a∈A
(b) It is symmetric i.e., a,b∈R⇒b,a∈R for all a,b∈A
(c) It is transitive i.e., a,b∈R and b,c∈R⇒a,c∈R for all a,b,c∈A.
It is not necessary that every relation which is symmetric, and transitive is also reflexive.
(ix) Inverse Relation: Let A, B be two sets and let R be a relation from a set A to a set B. Then, the inverse of R, denoted by R-1, is a relation from B to A and is defined by R-1=b,a:a,b∈R.
Clearly, a,b∈R⇔b,a∈R.
Also, DomainR=RangeR-1 and RangeR=DomainR-1.
10. Binary Operations
Let the set of numbers on which the binary operations are performed be X. The operations (addition, subtraction, division, multiplication etc.) can be generalized as a binary operation and are performed on two elements (Say a and b) from set X. The result of the operations on a and b is another element from the same set X.
Thus, the binary operation can be defined as an operation which is performed on a set A. The function is given by A*A→A.
So, the operation performed on operands a and b is denoted by a*b.
(i) Commutative binary operation
A binary operation ‘ *’ on a set S is said to be commutative binary operation, if a*b=b*a for all a,b∈S.
The binary operations addition + and multiplication × are commutative binary operations on Z. However, the binary operation subtraction – is not a commutative binary operation on Z as 3-2≠2-3.
(ii) Associative binary operation
A binary operation ‘ *’ on a set S is said to be an associative binary operation, if a*b*c=a*b*cfor all a,b∈S.
If S is a non-empty set, then union ∪ and intersection ∩ are both commutative and associative.
(iii) Distributive binary operation
Let S be a non-empty set and * and ‘ ⊙’ be two binary operations on S. Then, ‘ *’ is said to be distributive over ⊙, if a*b⊙c=a*b⊙a*c and b⊙c*a=b*a⊙c*a for all a,b,c∈Z.
However, addition + is not distributive over multiplication ⋅ because 2+3*5≠2+3*2+5.
(iv) Identity element
Let ‘ *’ be a binary operation on a set S. If there exists an element e∈S, such that
a*e=a=e*a for all a∈S.
Then, e is called an identity element for the binary operation ‘ *’ on set S.
Consider the binary operation of addition + on Z. We know that 0∈Z such that
a+0=a=0+a for all a∈Z.
So, ‘ 0’ is the identity element for addition on Z.
If we consider multiplication on Z, then 1 is the identity element for multiplication in Z, because
1×a=a=a×1 for all a∈Z.
(v) Invertible element:
Let ‘ *’ be a binary operation on a set S, and let e be the identity element in S for the binary operation * on S.
Then, an element a∈S is called an invertible element if there exists an element b∈S such that
a*b=e=b*a.
The element b is called an inverse of element a.
Consider the binary operation addition + on Z. Clearly, 0 is the identity element for addition on Z and for any integer a, we have a+-a=0=-a+a.
So, -a is the inverse of a∈Z.
Multiplication is also a binary operation on Z and 1 is the identity element for multiplication on Z. But, no element other than 1∈Z is invertible.