Sets/Relation/Function

Sets/Relation/Function

1.1 Introduction

 

  • The theory of sets was developed by German mathematician Georg Cantor.
  • Sets are used to define the concepts of relations and functions.
  • The study of geometry, sequences, probability, etc. requires the knowledge of sets.

1.2 Sets And Their Representations

  • A set is a well-defined collection of objects.
  • Sets are usually denoted by capital letters A, B, C, X, Y, Z, etc.
  • The elements of a set are represented by small letters a, b, c, x, y, z, etc.
1.2.1 Representation Of Sets
  • There are two methods of representing a set:
  • Roster or tabular form
  • Set – builder form.
1.2.1.1 Roster Form:
  • In roster form, all the elements of a set are listed, the elements are being separated by commas and are enclosed within braces { }.
  • Example: The set of all even positive integers less than 7 is described in roster form as { 2, 4, 6 }.
1.2.1.2 Set-Builder Form
  • In set-builder form, all the elements of a set possess a single common property which is not possessed by any element outside the set.
  • Example: In the set {a, e, i, o, u}, all the elements possess a common property, namely, each of them is a vowel in the English alphabet, and no other letter possess this property.
  • Denoting this set by K, we write,

K = { x : x is a vowel in English alphabet }

Example:
  • Write the set L = {1, 4, 9, 16, 25, . . . }in set-builder form.
Solution:
  • We may write the set L as

L = {x : x is the square of a natural number}

  • Alternatively, we can write

L = {x : x = n2 , where n  N}

1.3 The Empty Set 

  • A set which does not contain any element is called the empty set or the null set or the void set.
  • The empty set is denoted by the symbol  or { }.
Example:
  • D = {x : x2 = 4, x is odd}. Then D is the empty set because the equation x2 = 4 is not satisfied by any odd value of x.

1.4 Finite and Infinite Sets

Finite Sets
  • A set which is empty or consists of a definite number of elements is called finite.
Example:
  • Let W be the set of the days of the week. Then W is finite.
Infinite Sets
  • An infinite set is a set with an infinite number of elements.
Example:
  • The set of all integers, {….., -1, 0, 1, 2, …..}

1.5 Equal Sets 

 

  • Two sets A and B are said to be equal if they have exactly the same elements and we write A = B.
Example:
  • Let A be the set of prime numbers less than 6 and P the set of prime factors of 30.
  • Then A and P are equal, since 2, 3 and 5 are the only prime factors of 30 and also these are less than 6.

1.6 Subsets

  • A set P is said to be a subset of a set Q if every element of P is also an element of Q.
  • It is expressed as P  Q;The symbol  stands for ‘is a subset of’.
Example:
  • Let A = { a, e, i, o, u } and B = { a, b, c, d }. Is A a subset of B?
Solution:
  • No, A is not a subset of B as every element of A is not an element of B.

1.7 Power Set

  • The collection of all subsets of a set A is called the power set of A.
  •  It is denoted by P(A).
Example:
  • If A = { 1, 2 }, then P(A) = { , {1}, {2}, {1, 2} }
Note:

Cardinal Number :- The total no. of elements are known as Cardinal no.

  • If A is a set with n(A) = m, then it can be shown that n[P(A)] = 2m .

1.8 Universal Set

  • A Universal Set is the set of all elements under consideration, denoted by capital U.
  • All other sets are subsets of the universal set.
Example:
  • Given that U = {5, 6, 7, 8, 9, 10, 11, 12}, list the elements of the following sets.
  • A = {x : x is a factor of 60}
  • B = {x : x is a prime number}
Solution:
  • The elements of sets A and B can only be selected from the given universal set U.
  • A = {5, 6, 10, 12}
  • B = {5, 7, 11}

1.9 Venn Diagrams

  • A Venn diagram is a drawing in which circular areas represent groups of items which share a common property.
  • The drawing consists of two or more circles, each representing a specific group or set.

Example:
  • Given U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 }, A = { 1, 2, 5, 6 } and B = { 3, 9 }, draw a Venn diagram to represent these sets.
Solution:
  • Given
  • U = { 1, 2, 3, 4, 5, 6, 7, 8, 9 },
  • A = { 1, 2, 5, 6 } and
  • B = { 3, 9 }.

1.10 Operations on Sets 

  • There are some operations which when performed on two sets give rise to another set. They are:
  • Union of sets
  • Intersection of sets
  • Difference of sets
  • Complement of a set
1.10.1 Union of Sets:
  • The union of two sets A and B is the set C which consists of all those elements which are either in A or in B (including those which are in both).
  • The symbol ‘’ is used to denote the union.
  • Symbolically, we write A  B = {x : x  A or x  B}
  • The union of two sets can be represented by a Venn diagram as shown in figure.

Example:
  • Let A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Find A  B?
Solution:
  • We have A = { 2, 4, 6, 8 } and B = { 6, 8, 10, 12 }. Then A  B = { 2, 4, 6, 8, 10, 12 }
Properties of the Operation of Union:
  • Commutative law : A  B = B  A
  • Associative law :  (A  B) C = A  (B C)
  • Law of identity element : A   = A
  • Idempotent law : A  A = A
  • Law of U : U  A = U

 

1.10.2 Intersection of Sets:
  • The intersection of sets A and B is the set of all elements which are common to both A and B.
  • The symbol  is used to denote the intersection.
  • Symbolically we write it as A  B = {x : x  A and x  B}
  • The shaded portion in the figure indicates the intersection of A and B.

Properties of Operation of Intersection:
  • Commutative law: A  B = B  A
  • Associative law: (A  B) C = A  (B  C)
  • Law of  and U:   =
  • Idempotent law: A  A = A
  • Distributive law:
1.10.3 difference of Sets:-
  • The difference of the sets A and B in this order is the set of elements which belong to A but not to B.
  • Symbolically, we write A – B and read as “A minus B”.
Example:
  • Let A = { 1, 2, 3, 4, 5, 6 }, B = { 2, 4, 6, 8 }. Find A – B and B – A.
Solution:
  • A – B = { 1, 3, 5 }, since the elements 1, 3, 5 belong to A but not to B
  • B – A = { 8 }, since the element 8 belongs to B and not to A.

1.11 Complement of a Set

  • Let U be the universal set and A is a subset of U. Then the complement of A is the set of all elements of U which are not the elements of A.
  • Symbolically, we write A’ to denote the complement of A with respect to U. Thus,

A’ = {x : x  U and x  A}

Example:
  • Let U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 } and A = { 1, 3, 5, 7, 9 }. Find A’.
Solution:
  • 2, 4, 6, 8, 10 are the only elements of U which do not belong to A.
  • Hence A’ = { 2, 4, 6, 8, 10 }
Properties of Complement Sets:
  • Complement laws:
  • A  A’ = U
  • A  A’ =
  • De Morgan’s law:
  • (A  B)’ = A’  B’
  • (A  B)’ = A’  B’
  • Law of double complementation:
  • (A’)’ = A
  • Law of empty set and universal set
Formulae:

n(A  B ) = n(A) +  n(B)  –  n(A  B)

n(A  B  C) = n(A) + n(B) + n(C) – n(A  B) – n(B  C) – n(A  C) + (A  B  C)

Example:
  • In a school there are 20 teachers who teach mathematics or physics. Of these, 12 teach mathematics and 4 teach both physics and mathematics. How many teach physics?
Solution:
  • Let M denote the set of teachers who teach mathematics and P denote the set of teachers who teach physics.

n (M  P) = 20,

n(M) = 12

n(M  P) = 4

  • To find n(P)

n (M  P) = n(M) + n(P) – n(M  P)

20 = 12 + n(P) – 4

n(P) =12

 

Thank You:- Prof Rajeev Tripathi

 

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