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Class X - Maths: Sets






Contents
  • Sets
  • Naming Of Sets And Elements Of Set
  • Notation of a Set
  • Objects Form a Set
  • Example Problems
  • Elements of a Set
  • Example Problems on sets
  • Properties of Sets
  • examples using the properties of sets
  • Representation Of A Set
  • Statement form
  • Roster Form Or Tabular Form
  • Set Builder Form
  • Different Notations In Sets
  • Standard Sets of Numbers
  • Subset
  • Super Set
  • Proper Subset
  • Power Set
  • Universal Set
  • Pairs of Sets
  • equal sets
  • Equivalent Set
  • Disjoint Sets
  • Overlapping Sets
  • Operations on Sets
  • Union Of Sets
  • Properties Of The Operation Of Union
  • Intersection of Sets
  • examples to find intersection of two given sets
  • Properties Of The Operation Of Intersection
  • Difference of Two Sets
  • Examples To Find The Difference Of Two Sets
  • Complement Of A Set
  • Some Properties Of Complement Sets
  • Cardinal Number Of A Set
  • Examples On Cardinal Number Of A Set
  • Cardinal Properties of Sets
  • Problems on Cardinal Properties of Sets
  • Venn Diagrams
  • Venn diagrams in different situations

Sets

What is set (in mathematics)?
➢ The collection of well-defined distinct objects is known as a set. The word well-defined refers to a specific property which makes it easy to identify whether the given object belongs to the set or not. The word 'distinct' means that the objects of a set must be all different.

For Example

  1. The collection of children in class VII whose weight exceeds 35 kg represents a set.
  2. The collection of all the intelligent children in class VII does not represent a set because the word intelligent is vague. What may appear intelligent to one person may not appear the same to another person.

Naming Of Sets And Elements Of Set

➢ The different objects that form a set are called the elements of a set. The elements of the set are written in any order and are not repeated. Elements are denoted by small letters.


Notation of a Set

➢ A set is usually denoted by capital letters and elements are denoted by small letters
If x is an element of set A, then we say x ϵ A. [x belongs to A]
If x is not an element of set A, then we say x ∉ A. [x does not belong to A]


For example:

The collection of vowels in the English alphabet.

Solution
Let us denote the set by V, then the elements of the set are a, e, i, o, u or we can say,
V = {a, e, i, o, u}.
We say a ∈ V, e ∈ V, i ∈ V, o ∈ V and u ∈ V.
Also, we can say b ∉ V, c ∉ v, d ∉ v, etc.


Objects Form a Set

How to state that whether the objects form a set or not?

➢ A collection of 'lovely flowers' is not a set, because the objects (flowers) to be included are not well-defined.

Reason: The word "lovely" is a relative term. What may appear lovely to one person may not be so to the other person.

➢ A collection of "Yellow flowers"is a set, because every red flowers will be included in this set i.e., the objects of the set are well-defined.
➢ A group of "Young singers"is not a set, as the range of the ages of young singers is not given and so it can't be decided that which singer is to be considered young i.e., the objects are not well-defined.
➢ A group of "Players with ages between 18 years and 25 years" is a set, because the range of ages of the player is given and so it can easily be decided that which player is to be included and which is to be excluded. Hence, the objects are well-defined.
➢ Now we will learn to state which of the following collections are set.


State, Giving Reason, Whether The Following Objects Form A Set Or Not:

(i) All problems of this book, which are difficult to solve.
Solution:

The given objects do not form a set.
Reason: Some problems may be difficult for one person but may not be difficult for some other persons, that is, the given objects are not well-defined.
Hence, they do not form a set.


(ii) All problems of this book, which are difficult to solve for Aaron.
Solution:

The given objects form a set.
Reason: It can easily be found that which are difficult to solve for Aaron and which are not difficult to solve for him.
Hence, the objects form a set.


(iii) All the objects heavier than 28 kg.
Solution:

The given objects form a set.

Reason: Every object can be compared, in weight, with 28 kg. Then it is very easy to select objects which are heavier than 28 kg i.e., the objects are well-defined.
Hence, the objects form a set.

The members (objects) of each of the following collections form a set:
➢ students in a class-room
➢ books in your school-bag
➢ counting numbers between 5 to 15
➢ students of your class, which are taller than you and so on.


Elements of a Set

What are the elements of a set or members of a set?
➢ The objects used to form a set are called its element or its members.
➢ Generally, the elements of a set are written inside a pair of curly (idle) braces and are represented by commas. The name of the set is always written in capital letter.


Solved Examples to find the elements or members of a set:


1. A = {v, w, x, y, z}
Here 'A' is the name of the set whose elements (members) are v, w, x, y, z.

2. If a set A = {3, 6, 9, 10, 13, 18}. State whether the following statements are 'true' or 'false':
(i) 7 ∈ A
(ii) 10 ∉ A
(iii) 13 ∈ A
(iv) 9, 10 ∈ A
(v) 10, 13, 14 ∈ A
Solution:

(i) 7 ∈ A
➢ False, since the element 7 does not belongs to the given set A.
(ii) 10 ∉ A
➢ False, since the element 10 belongs to the given set A.
(iii) 13 ∈ A
➢ True, since the element 13 belongs to the given set A.
(iv) 9, 10 ∈ A
➢ True, since the elements 9 and 10 both belong to the given set A.
(v) 10, 13, 14 ∈ A
➢ False, since the element 14 does not belongs to the given set A.


3. If set Z = {2,4, 6, 8, 10, 12, 14}. State which of the following statements are 'correct' and which are 'wrong' along with the correct explanations
(i) 5 ∈ Z
(ii) 12 ∈ Z
(iii) 14 ∈ Z
(iv) 9∈ Z
(v) Z is a set of even numbers between 2 and 16.
(vi) 4, 6 and 10 are members of the set Z.

Solution:
(i) 5 ∈ Z
Wrong, since 5 does not belongs to the given set Z i.e. 5 ∉ Z
(ii) 12 ∈ Z
Correct, since 12 belongs to the given set Z.
(iii) 14 ∈ Z
Correct, since 14 belongs to the given set Z.
(iv) 9∈ Z
Wrong, since 9 does not belongs to the given set Z i.e. 9 ∉ Z
(v) Z is a set of even numbers between 2 and 16.
Correct, since the elements of the set Z consists of all the multiples of 2 between 2 and 16.
(vi) 4, 6 and 10 are members of the set Z.
Correct, since the 4, 6 and 10 those numbers belongs to the given set Z.


Properties of Sets

1. The change in order of writing the elements does not make any changes in the set.
➢ In other words the order in which the elements of a set are written is not important. Thus, the set {a, b, c} can also be written as {a, c, b} or {b, c, a} or {b, a, c} or {c, a, b} or {c, b, a}.
For Example:

➢ Set A = {4, 6, 7, 8, 9} is same as set A = {8, 4, 9, 7, 6}
i.e., {4, 6, 7, 8, 9} = {8, 4, 9, 7, 6}
➢ Similarly, {w, x, y, z} = {x, z, w, y} = {z, w, x, y} and so on.


2. If one or many elements of a set are repeated, the set remains the same.
➢ In other words the elements of a set should be distinct. So, if any element of a set is repeated number of times in the set, we consider it as a single element.
Thus, {1, 1, 2, 2, 3, 3, 4, 4, 4} = {1, 2, 3, 4}
➢ The set of letters in the word 'GOOGLE' = {G, O, L, E}
Example:
➢ The set A = {5, 6, 7, 6, 8, 5, 9} is same as set A= {5, 6, 7, 8, 9}
i.e., {5, 6, 7, 6, 8, 5, 9} = {5, 6, 7, 8, 9}
➢ In general, the elements of a set are not repeated. Thus,
➢ if T is a set of letters of the word 'moon': then T = {m, o, n},
➢ There are two o's in the word 'moon' but it is written in the set only once.
➢ if U = {letters of the word 'COMMITTEE'};
then U = {C, O, M, T, E}


Solved examples using the properties of sets

1. Write the set of vowels used in the word 'UNIVERSITY'.
Solution:

Set V = {U, I, E}


2. For each statement, given below, state whether it is true or false along with the explanations.

(i) {9, 9, 9, 9, 9, ........} = {9}
(ii) {p, q, r, s, t} = {t, s, r, q, p}

Solution:

(i) {9, 9, 9, 9, 9, ........} = {9}
True, since repetition of elements does not change the set.
(ii) {p, q, r, s, t} = {t, s, r, q, p}
True, since the change in order of writing the elements does not change the set.


Representation Of A Set

In representation of a set the following three methods are commonly used:
(i) Statement form method
(ii) Roster or tabular form method
(iii) Rule or set builder form method


1. Statement form

➢ In this, well-defined description of the elements of the set is given and the same are enclosed in curly brackets.

For example

(i) The set of odd numbers less than 7 is written as: {odd numbers less than 7}.
(ii) A set of football players with ages between 22 years to 30 years.
(iii) A set of numbers greater than 30 and smaller than 55.
(iv) A set of students in class VII whose weights are more than your weight.


2. Roster Form Or Tabular Form

➢ In this, elements of the set are listed within the pair of brackets { } and are separated by commas.

For Example

(i) Let N denote the set of first five natural numbers.
Therefore, N = {1, 2, 3, 4, 5} → Roster Form
(ii) The set of all vowels of the English alphabet.
Therefore, V = {a, e, i, o, u} → Roster Form
(iii) The set of all odd numbers less than 9.
Therefore, X = {1, 3, 5, 7} → Roster Form
(iv) The set of all natural number which divide 12.
Therefore, Y = {1, 2, 3, 4, 6, 12} → Roster Form
(v) The set of all letters in the word MATHEMATICS.
Therefore, Z = {M, A, T, H, E, I, C, S} → Roster Form
(vi) W is the set of last four months of the year.
Therefore, W = {September, October, November, December} → Roster Form

Note:
The order in which elements are listed is immaterial but elements must not be repeated.


3. Set Builder Form

➢ In this, a rule, or the formula or the statement is written within the pair of brackets so that the set is well defined. In the set builder form, all the elements of the set, must possess a single property to become the member of that set.

In this form of representation of a set, the element of the set is described by using a symbol 'x' or any other variable followed by a colon.
➢ The symbol ':' or '|' is used to denote such that and then we write the property possessed by the elements of the set and enclose the whole description in braces. In this, the colon stands for 'such that' and braces stand for 'set of all'.

For Example

(i) Let P is a set of counting numbers greater than 12;
the set P in set-builder form is written as :
P = {x : x is a counting number and greater than 12}
or
P = {x | x is a counting number and greater than 12}
➢ This will be read as, 'P is the set of elements x such that x is a counting number and is greater than 12'.

Note:
The symbol ':' or '|' placed between 2 x's stands for such that.
(ii) Let A denote the set of even numbers between 6 and 14. It can be written in the set builder form as;
A = {x|x is an even number, 6 < x < 14}
or A = {x : x ∈ P, 6 < x < 14 and P is an even number}
(iii) If X = {4, 5, 6, 7} . This is expressed in roster form.
Let us express in set builder form.
X = {x : x is a natural number and 3 < x < 8}
(iv) The set A of all odd natural numbers can be written as
A = {x : x is a natural number and x = 2n + 1 for n ∈ W}


Solved Example Using The Three Methods Of Representation Of A Set

➢ The set of integers lying between -2 and 3.
Statement form: {I is a set of integers lying between -2 and 3}
Roster form: I = {-1, 0, 1, 2}
Set builder form: I = {x : x ∈ I, -2 < x < 3}


Different Notations In Sets

➢ To learn about sets we shall use some accepted notations for the familiar sets of numbers.

Some Of The Different Notations Used In Sets Are:

These are the different notations in sets generally required while solving various types of problems on sets.

Note

➢ The pair of curly braces { } denotes a set. The elements of set are written inside a pair of curly braces separated by commas.

  • The set is always represented by a capital letter such as; A, B, C, ........ .
  • If the elements of the sets are alphabets then these elements are written in small letters.
  • The elements of a set may be written in any order.
  • The elements of a set must not be repeated.
  • The Greek letter Epsilon '∈' is used for the words 'belongs to', 'is an element of', etc.
  • Therefore, x ∈ A will be read as 'x belongs to set A' or 'x is an element of the set A'.
  • The symbol '∉' stands for 'does not belongs to' also for 'is not an element of'.
  • Therefore, x ∉ A will



Standard Sets of Numbers

1. N = Natural Numbers

= Set of all numbers starting from 1
= {Set of all numbers 1, 2, 3, .....} → Statement form
= {1, 2, 3, .......} → Roster form
= {x :x is a counting number starting from 1} → Set builder form
Therefore, the set of natural numbers is denoted by N i.e., N = {1, 2, 3, .......}


2. W = Whole numbers

= Set containing zero and all natural numbers → Statement form
= {0, 1, 2, 3, .......} → Roster form
= {x :x is a zero and all natural numbers} → Set builder form
Therefore, the set of whole numbers is denoted by W i.e., W = {0, 1, 2, .......}


3. Z or I = Integers

= Set containing negative of natural numbers, zero and the natural numbers
→ Statement form
= {........., -3, -2, -1, 0, 1, 2, 3, .......} → Roster form
= {x :x is a containing negative of natural numbers, zero and the natural numbers}
→ Set builder form
Therefore, the set of integers is denoted by I or Z i.e., I = {...., -2, -1, 0, 1, 2, ....}


4. E = Even natural numbers.

= Set of natural numbers, which are divisible by 2 → Statement form
= {2, 4, 6, 8, ..........} → Roster form
= {x :x is a natural number, which are divisible by 2} → Set builder form
Therefore, the set of even natural numbers is denoted by E i.e., E = {2, 4, 6, 8,.......}


5. O = Odd natural numbers.

= Set of natural numbers, which are not divisible by 2 → Statement form
= {1, 3, 5, 7, 9, ..........} → Roster form
= {x :x is a natural number, which are not divisible by 2} → Set builder form

Subset

Definition of Subset
➢ If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A ⊆ B or B ⊇ A
The symbol ⊂ stands for 'is a subset of' or 'is contained in'

  • Every set is a subset of itself, i.e., A ⊂ A, B ⊂ B.
  • Empty set is a subset of every set.
  • Symbol '⊆' is used to denote 'is a subset of' or 'is contained in'.
  • A ⊆ B means A is a subset of B or B contains A.
  • B ⊆ A means B is a subset of A or A containes B.
For example

1.Let A = {2, 4, 6}
B = {6, 4, 8, 2}
Here A is a subset of B
Since, all the elements of set A are contained in set B.
But B is not the subset of A
Since, all the elements of set B are not contained in set A.
if A`sub`B And B`sub`A,then A = B they are equal sets.Every set is a subset of itself


1. The set N of natural numbers is a subset of the set Z of integers and we write N ⊂ Z.
2. Let A = {2, 4, 6}

B = {x : x is an even natural number less than 8}
Here A ⊂ B and B ⊂ A.
Hence, we can say A = B

3. Let A = {1, 2, 3, 4}

B = {4, 5, 6, 7}
Here A ⊄ B and also B ⊄ C
[⊄ denotes 'not a subset of']


Super Set

➢ Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote 'is a super set of'

For example

A = {a, e, i, o, u}
B = {a, b, c, ............., z}
Here A ⊆ B i.e., A is a subset of B but B ⊇ A i.e., B is a super set of A


Proper Subset

➢ If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B.
The symbol '⊂' is used to denote proper subset. Symbolically, we write A⊂ B.

For example

1.A = {1, 2, 3, 4}
Here n(A) = 4
B = {1, 2, 3, 4, 5}
Here n(B) = 5

➢ We observe that, all the elements of A are present in B but the element '5' of B is not present in A.
➢ So, we say that A is a proper subset of B.
Symbolically, we write it as A ⊂ B

Note

No set is proper set of itself.Null set or Ø is a proper subset of every set


2. A = {p, q, r}
B = {p, q, r, s, t}
➢ Here A is a proper subset of B as all the elements of set A are in set B and also A ≠ B.

Note

No set is proper set of itself.Null set or empty set is a proper subset of every set


Power Set

➢ The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.
For example

➢ If A = {p, q} then all the subsets of A will be
P(A) = {∅, {p}, {q}, {p, q}}
Number of elements of P(A) = n[P(A)] = 4
In general, n[P(A)] = 2m where m is the number of elements in set A.


Universal Set

➢ A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is ∪ or ξ.

For example

1. If A = {1, 2, 3} B = {2, 3, 4} C = {3, 5, 7}
then U = {1, 2, 3, 4, 5, 7}
[Here A ⊆ U, B ⊆ U, C ⊆ U and U ⊇ A, U ⊇ B, U ⊇ C]


2. If P is a set of all whole numbers and Q is a set of all negative numbers then the universal set is a set of all integers.

3. If A = {a, b, c} B = {d, e} C = {f, g, h, i}
then U = {a, b, c, d, e, f, g, h, i} can be taken as universal set.


Pairs of Sets

Equal Set

➢ Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa.
The symbol to denote an equal set is =.
A = B means set A is equal to set B and set B is equal to set A.

For Example

A = {2, 3, 5}
B = {5, 2, 3}
Here, set A and set B are equal sets.


Equivalent Set

➢ Two sets A and B are said to be equivalent sets if they contain the same number of elements.
The symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.

For example

A = {p, q, r}
B = {2, 3, 4}
➢ Here, we observe that both the sets contain three elements.

Note

equal sets are always equivalent.
equivalent sets may not be equal


Disjoint Sets

➢ Two sets A and B are said to be disjoint, if they do not have any element in common.

For example

A = {x : x is a prime number}
B = {x : x is a composite number}.
Clearly, A and B do not have any element in common and are disjoint sets.


Overlapping Sets

➢ Two sets A and B are said to be overlapping if they contain at least one element in common.

For Example

➢ A = {a, b, c, d}
B = {a, e, i, o, u}
X = {x : x ∈ N, x < 4}
Y = {x : x ∈ I, -1 < x < 4}
Here, the two sets contain three elements in common, i.e., (1, 2, 3)


Operations on Sets

➢ The four basic operations on sets

Solution:

The four basic operations are:

  1. Union of Sets
  2. Intersection of sets
  3. Complement of the Set
  4. Cartesian Product of sets

Union Of Sets

➢ Definition of Union of Sets:

➢ To find the union of two given sets A and B is a set which consists of all the elements of A and all the elements of B such that no element is repeated.
➢ The symbol for denoting union of sets is '∪'.

For Example

➢ Let set A = {2, 4, 5, 6}
and set B = {4, 6, 7, 8}

➢ Taking every element of both the sets A and B, without repeating any element, we get a new set = {2, 4, 5, 6, 7, 8}

➢ This new set contains all the elements of set A and all the elements of set B with no repetition of elements and is named as union of set A and B.
➢ The symbol used for the union of two sets is '∪'.
➢ Therefore, symbolically, we write union of the two sets A and B is A ∪ B which means A union B.
Therefore, A ∪ B = {x : x ∈ A or x ∈ B}


Solved Examples To Find Union Of Two Given Sets

1. If A = {1, 3, 7, 5} and B = {3, 7, 8, 9}. Find union of two set A and B.
➢ Solution:

A ∪ B = {1, 3, 5, 7, 8, 9}
No element is repeated in the union of two sets. The common elements 3, 7 are taken only once.


2. Let X = {a, e, i, o, u} and Y = {Φ}. Find union of two given sets X and Y.
➢ Solution:

X ∪ Y = {a, e, i, o, u}
Therefore, union of any set with an empty set is the set itself.


3. If set P = {2, 3, 4, 5, 6, 7}, set Q = {0, 3, 6, 9, 12} and set R = {2, 4, 6, 8}.
(i) Find the union of sets P and Q
(ii) Find the union of two set P and R
(iii) Find the union of the given sets Q and R

Solution

➢ (i) Union of sets P and Q is P ∪ Q
The smallest set which contains all the elements of set P and all the elements of set Q is {0, 2, 3, 4, 5, 6, 7, 9, 12}.
➢ (ii) Union of two set P and R is P ∪ R
The smallest set which contains all the elements of set P and all the elements of set R is {2, 3, 4, 5, 6, 7, 8}.
➢ (iii) Union of the given sets Q and R is Q ∪ R
The smallest set which contains all the elements of set Q and all the elements of set R is {0, 2, 3, 4, 6, 8, 9, 12}.

Notes:
➢ A and B are the subsets of A ∪ B
➢ The union of sets is commutative, i.e., A ∪ B = B ∪ A.
➢ The operations are performed when the sets are expressed in roster form.


Some Properties Of The Operation Of Union

(i) A∪B = B∪A  (Commutative law)
(ii) A∪(B∪C) = (A∪B)∪C  (Associative law)
(iii) A ∪Φ = A  (Law of identity element, is the identity of ∪)
(iv) A∪A = A  (Idempotent law)
(v) U∪A = U  (Law of ∪) ∪ is the universal set.

Intersection of Sets

➢ To find the intersection of two given sets A and B is a set which consists of all the elements which are common to both A and B.
The symbol for denoting intersection of sets is '∩'.

For Example

Let set A = {2, 3, 4, 5, 6}
and set B = {3, 5, 7, 9}

➢ In this two sets, the elements 3 and 5 are common. The set containing these common elements i.e., {3, 5} is the intersection of set A and B.
➢ The symbol used for the intersection of two sets is '∩'.
➢ Therefore, symbolically, we write intersection of the two sets A and B is A∩ B which means A intersection B.
➢ The intersection of two sets A and B is represented as A ∩ B = {x : x ∈ A and x ∈ B}


Solved examples to find intersection of two given sets:

1. If A = {2, 4, 6, 8, 10} and B = {1, 3, 8, 4, 6}. Find intersection of two set A and B.
Solution:

A ∩ B = {4, 6, 8}
Therefore, 4, 6 and 8 are the common elements in both the sets.


2. If X = {a, b, c} and Y = {Φ}. Find intersection of two given sets X and Y.
Solution:

X ∩ Y = { }


3. If set A = {4, 6, 8, 10, 12}, set B = {3, 6, 9, 12, 15, 18} and set C = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}.

(i) Find the intersection of sets A and B.
(ii) Find the intersection of two set B and C.
(iii) Find the intersection of the given sets A and C.
Solution:

➢ (i) Intersection of sets A and B is A ∩ B
Set of all the elements which are common to both set A and set B is {6, 12}.
➢ (ii) Intersection of two set B and C is B ∩ C
Set of all the elements which are common to both set B and set C is {3, 6, 9}.
➢ (iii) Intersection of the given sets A and C is A ∩ C
Set of all the elements which are common to both set A and set C is {4, 6, 8, 10}.

Note:

A ∩ B subset of A and B.
Intersection Of A Set Is commutative.
A ∩ B = B ∩ A

➢ Operations are performed when the set is expressed in the roster form.


Some Properties Of The Operation Of Intersection

(i) A∩B = B∩A (Commutative law)
(ii) (A∩B)∩C = A∩ (B∩C) (Associative law)
(iii) Φ∩ A = Φ (Law of Φ)
(iv) U∩A = A (Law of ∪)
(v) A∩A = A (Idempotent law)
(vi) A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪
Also, A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩


Difference of Two Sets

➢ find the difference of two sets
➢ If A and B are two sets, then their difference is given by A - B or B - A.
If A = {2, 3, 4} and B = {4, 5, 6}
A - B means elements of A which are not the elements of B.
i.e., in the above example A - B = {2, 3}
In general, B - A = {x : x ∈ B, and x ∉ A}
If A and B are disjoint sets, then A - B = A and B - A = B


Solved Examples To Find The Difference Of Two Sets

1. A = {1, 2, 3} and B = {4, 5, 6}.
Find the difference between the two sets:
(i) A and B
(ii) B and A

Solution:
The two sets are disjoint as they do not have any elements in common.
(i) A - B = {1, 2, 3} = A
(ii) B - A = {4, 5, 6} = B


2. Let A = {a, b, c, d, e, f} and B = {b, d, f, g}.

Find the difference between the two sets:
(i) A and B
(ii) B and A

Solution:

(i) A - B = {a, c, e}
Therefore, the elements a, c, e belong to A but not to B

(ii) B - A = {g)
Therefore, the element g belongs to B but not A.


3. Given three sets P, Q and R such that:
P = {x : x is a natural number between 10 and 16},
Q = {y : y is a even number between 8 and 20} and
R = {7, 9, 11, 14, 18, 20}
(i) Find the difference of two sets P and Q
(ii) Find Q - R
(iii) Find R - P
(iv) Find Q - P

Solution:

According to the given statements:
P = {11, 12, 13, 14, 15}
Q = {10, 12, 14, 16, 18}
R = {7, 9, 11, 14, 18, 20}

(i) P - Q = {Those elements of set P which are not in set Q}
= {11, 13, 15}
(ii) Q - R = {Those elements of set Q not belonging to set R}
= {10, 12, 16}
(iii) R - P = {Those elements of set R which are not in set P}
= {7, 9, 18, 20}
(iv) Q - P = {Those elements of set Q not belonging to set P}
= {10, 16, 18}


Complement Of A Set

For Example
If ξ = {1, 2, 3, 4, 5, 6, 7}
A = {1, 3, 7} find A'.

Solution

We observe that 2, 4, 5, 6 are the only elements of ξ which do not belong to A.
Therefore, A' = {2, 4, 5, 6}

Note:

The compliment of a universal set is an empty set.
The compliment of an empty set a universal set.
The set and its compliment are disjoint sets.


For Example

➢ 1.Let the set of natural numbers be the universal set and A is a set of even natural numbers,
➢ then A' {x: x is a set of odd natural numbers}
➢ 2. Let ξ = The set of letters in the English alphabet.
➢ A = The set of consonants in the English alphabet
then A' = The set of vowels in the English alphabet.


3. Show that;

(a) The Complement Of A Universal Set Is An Empty Set.

➢ Let ξ denote the universal set, then
ξ' = The set of those elements which are not in ξ.
= empty set = Φ
Therefore, ξ = Φ so the complement of a universal set is an empty set.


(b) A Set And Its Complement Are Disjoint Sets.

➢ Let A be any set then A' = set of those elements of ξ which are not in A.
Let x ∉ A, then x is an element of ξ not contained in A
So x ∈ A'
➢ Therefore, A and A' are disjoint sets.
Therefore, Set and its complement are disjoint sets
➢ Similarly, in complement of a set when U be the universal set and A is a subset of U. Then the complement of A is the set 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}
➢ Obviously A' = {U - A}

For Example; Let U = {2, 4, 6, 8, 10, 12, 14, 16}
A = {6, 10, 4, 16}
A' = {2, 8, 12, 14}
We observe that 2, 8, 12, 14 are the only elements of U which do not belong to A.


Some Properties Of Complement Sets

(i) A ∪ A' = A' ∪ A = ∪ (Complement law)
(ii) (A ∩ A') = Φ (Complement law)
(iii) (A ∪ B)' = A' ∩ B' (De Morgan's law)
(iv) (A ∩ B)' = A' ∪ B' (De Morgan's law)
(v) (A')' = A (Law of complementation)
(vi) Φ' = ∪ (Law of empty set )
(vii) ∪' = Φ and universal set)

Cardinal Number Of A Set

➢ The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and read as 'the number of elements of the set'.

For Example

(i) Set A = {2, 4, 5, 9, 15} has 5 elements.
Therefore, the cardinal number of set A = 5. So, it is denoted as n(A) = 5.

(ii) Set B = {w, x, y, z} has 4 elements.
Therefore, the cardinal number of set B = 4. So, it is denoted as n(B) = 4.

(iii) Set C = {Florida, New York, California} has 3 elements.
Therefore, the cardinal number of set C = 3. So, it is denoted as n(C) = 3.

(iv) Set D = {3, 3, 5, 6, 7, 7, 9} has 5 element.
Therefore, the cardinal number of set D = 5. So, it is denoted as n(D) = 5.

(v) Set E = { } has no element.
Therefore, the cardinal number of set D = 0. So, it is denoted as n(D) = 0.

Note:

(i) Cardinal number of an infinite set is not defined.
(ii) Cardinal number of empty set is 0 because it has no element.


Solved Examples On Cardinal Number Of A Set

1. Write the cardinal number of each of the following sets:

(i) X = {letters in the word MALAYALAM}
(ii) Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
(iii) Z = {natural numbers between 20 and 50, which are divisible by 7}

Solution

(i) Given, X = {letters in the word MALAYALAM}
Then, X = {M, A, L, Y}
Therefore, cardinal number of set X = 4, i.e., n(X) = 4

(ii) Given, Y = {5, 6, 6, 7, 11, 6, 13, 11, 8}
Then, Y = {5, 6, 7, 11, 13, 8}
Therefore, cardinal number of set Y = 6, i.e., n(Y) = 6

(iii) Given, Z = {natural numbers between 20 and 50, which are divisible by 7}
Then, Z = {21, 28, 35, 42, 49}
Therefore, cardinal number of set Z = 5, i.e., n(Z) = 5


2. Find the cardinal number of a set from each of the following:
(i) P = {x | x ∈ N and x < 6}
(ii) Q = {x | x is a factor of 20}

Solution

(i) Given, P = {x | x ∈ N and x2 < 30}
Then, P = {1, 2, 3, 4, 5}
Therefore, cardinal number of set P = 5, i.e., n(P) = 5

(ii) Given, Q = {x | x is a factor of 20}
Then, Q = {1, 2, 4, 5, 10, 20}
Therefore, cardinal number of set Q = 6, i.e., n(Q) = 6


Cardinal Properties of Sets

➢ We have already learnt about the union, intersection and difference of sets. Now, we will go through some practical problems on sets related to everyday life.

If A and B are finite sets, then

  • n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
    If A ∩ B = Φ , then n(A ∪ B) = n(A) + n(B)
    It is also clear from the Venn diagram that
  • n(A - B) = n(A) - n(A ∩ B)
  • n(B - A) = n(B) - n(A ∩ B)
realnumbers

Problems on Cardinal Properties of Sets

1. If P and Q are two sets such that P ∪ Q has 40 elements, P has 22 elements and Q has 28 elements, how many elements does P ∩ Q have?
Solution:

Given n(P ∪ Q) = 40, n(P) = 18, n(Q) = 22
We know that n(P U Q) = n(P) + n(Q) - n(P ∩ Q)
So, 40 = 22 + 28 - n(P ∩ Q)
40 = 50 - n(P ∩ Q)
Therefore, n(P ∩ Q) = 50 - 40
= 10


2. In a class of 40 students, 15 like to play cricket and football and 20 like to play cricket. How many like to play football only but not cricket?
Solution:

Let C = Students who like cricket
F = Students who like football
C ∩ F = Students who like cricket and football both
C - F = Students who like cricket only
F - C = Students who like football only.
n(C) = 20 , n(C ∩ F) = 15 , n (C U F) = 40, n (F) = ?
n(C ∪ F) = n(C) + n(F) - n(C ∩ F)
40 = 20 + n(F) - 15
40 = 5 + n(F)
40 - 5 = n(F)
Therefore, n(F)= 35
Therefore, n(F - C) = n(F) - n (C ∩ F)
= 35 - 15
= 20
Therefore, Number of students who like football only but not cricket = 20


More problems on cardinal properties of sets

3. There is a group of 80 persons who can drive scooter or car or both. Out of these, 35 can drive scooter and 60 can drive car. Find how many can drive both scooter and car? How many can drive scooter only? How many can drive car only?
Solution:

Let S = {Persons who drive scooter}
C = {Persons who drive car}
Given, n(S ∪ C) = 80 n(S) = 35 n(C) = 60
Therefore, n(S ∪ C) = n(S) + n(C) - n(S ∩ C)
80 = 35 + 60 - n(S ∩ C)
80 = 95 - n(S ∩ C)
Therefore, n(S∩C) = 95 - 80 = 15
Therefore, 15 persons drive both scooter and car.
Therefore, the number of persons who drive a scooter only = n(S) - n(S ∩ C)
= 35 - 15
= 20
Also, the number of persons who drive car only = n(C) - n(S ∩ C)
= 60 - 15
= 45


4. It was found that out of 45 girls, 10 joined singing but not dancing and 24 joined singing. How many joined dancing but not singing? How many joined both?
Solution:

Let S = {Girls who joined singing}
D = {Girls who joined dancing}
Number of girls who joined dancing but not singing = Total number of girls - Number of girls who joined singing
45 - 24
= 21
Now, n(S - D) = 10 n(S) =24
Therefore, n(S - D) = n(S) - n(S ∩ D)
⇒ n(S ∩ D) = n(S) - n(S - D)
= 24 - 10
= 14
Therefore, number of girls who joined both singing and dancing is 14.


Venn Diagrams

➢ Venn diagrams are useful in solving simple logical problems. Let us study about them in detail.
Mathematician John Venn introduced the concept of representing the sets pictorially by means of closed geometrical figures called Venn diagrams.
➢ In Venn diagrams, the Universal Set ξ is represented by a rectangle and all other sets under consideration by circles within the rectangle. In this chapter, we will use Venn diagrams to illustrate various operations (union, intersection, difference).

Venn Diagrams

➢ Pictorial representations of sets represented by closed figures are called set diagrams or Venn diagrams.
➢ Venn diagrams are used to illustrate various operations like union, intersection and difference.

We can express the relationship among sets through this in a more significant way.

In this,

  • A rectangle is used to represent a universal set.
  • Circles or ovals are used to represent other subsets of the universal set.

➢ In this diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the rectangle. We represented disjoint set by disjoint circles and intersecting sets by intersecting circles.

Venn diagrams in different situations
  • If a set A is a subset of set B, then the circle representing set A is drawn inside the circle representing set B.
  • realnumbers
  • If set A and set B have some elements in common, then to represent them, we draw two circles which are overlapping.
  • realnumbers
  • If set A and set B are disjoint, then they are represented by two non-intersecting circles.
  • realnumbers

➢ In this diagrams, the universal set is represented by a rectangular region and its subsets by circles inside the rectangle. We represented disjoint set by disjoint circles and intersecting sets by intersecting circles.

Formulae

(1).What is set (in mathematics)?
The collection of well-defined distinct objects is known as a set.


A set is usually denoted by capital letters and elements are denoted by small letters
If x is an element of set A, then we say x ϵ A. [x belongs to A]


What are the elements of a set or members of a set?
The objects used to form a set are called its element or its members.


Statement form
In this, well-defined description of the elements of the set is given and the same are enclosed in curly brackets.


Roster Form Or Tabular Form
In this, elements of the set are listed within the pair of brackets { } and are separated by commas.


Set Builder Form
In this, a rule, or the formula or the statement is written within the pair of brackets so that the set is well defined. In the set builder form, all the elements of the set, must possess a single property to become the member of that set.


Standard Sets of Numbers
N = Natural Numbers
= Set of all numbers starting from 1


W = Whole numbers
= Set containing zero and all natural numbers → Statement form


Z or I = Integers
= Set containing negative of natural numbers, zero and the natural numbers
→ Statement form
= {........., -3, -2, -1, 0, 1, 2, 3, .......} → Roster form


E = Even natural numbers.
= Set of natural numbers, which are divisible by 2 → Statement form
= {2, 4, 6, 8, ..........} → Roster form


O = Odd natural numbers.
= Set of natural numbers, which are not divisible by 2 → Statement form
= {1, 3, 5, 7, 9, ..........} → Roster form


Definition of Subset
➢ If A and B are two sets, and every element of set A is also an element of set B, then A is called a subset of B and we write it as A ⊆ B or B ⊇ A
The symbol ⊂ stands for 'is a subset of' or 'is contained in'


superset
➢ Whenever a set A is a subset of set B, we say the B is a superset of A and we write, B ⊇ A.
Symbol ⊇ is used to denote 'is a super set of'


Proper Subset
➢ If A and B are two sets, then A is called the proper subset of B if A ⊆ B but B ⊇ A i.e., A ≠ B.
The symbol '⊂' is used to denote proper subset. Symbolically, we write A⊂ B.


Power Set
➢ The collection of all subsets of set A is called the power set of A. It is denoted by P(A). In P(A), every element is a set.


Universal Set
➢ A set which contains all the elements of other given sets is called a universal set. The symbol for denoting a universal set is ∪ or ξ.


Equal Set
Two sets A and B are said to be equal if all the elements of set A are in set B and vice versa.
The symbol to denote an equal set is =.


Equivalent Set
➢ Two sets A and B are said to be equivalent sets if they contain the same number of elements.
The symbol to denote equivalent set is ↔.
A ↔ means set A and set B contain the same number of elements.


Disjoint Sets
➢ Two sets A and B are said to be disjoint, if they do not have any element in common.


Overlapping Sets
Two sets A and B are said to be overlapping if they contain at least one element in common.


Some Properties Of The Operation Of Union
(i) A∪B = B∪A  (Commutative law)
(ii) A∪(B∪C) = (A∪B)∪C  (Associative law)
(iii) A ∪Φ = A  (Law of identity element, is the identity of ∪)
(iv) A∪A = A  (Idempotent law)
(v) U∪A = U  (Law of ∪) ∪ is the universal set.


intersection of two sets
To find the intersection of two given sets A and B is a set which consists of all the elements which are common to both A and B.
The symbol for denoting intersection of sets is '∩'.


(i) A∩B = B∩A (Commutative law)
(ii) (A∩B)∩C = A∩ (B∩C) (Associative law)
(iii) Φ∩ A = Φ (Law of Φ)
(iv) U∩A = A (Law of ∪)
(v) A∩A = A (Idempotent law)
(vi) A∩(B∪C) = (A∩B) ∪ (A∩C) (Distributive law) Here ∩ distributes over ∪
Also, A∪(B∩C) = (AUB) ∩ (AUC) (Distributive law) Here ∪ distributes over ∩


find the difference of two sets
➢ If A and B are two sets, then their difference is given by A - B or B - A.


The Complement Of A Universal Set Is An Empty Set.
➢ Let ξ denote the universal set, then
ξ' = The set of those elements which are not in ξ.
= empty set = Φ
Therefore, ξ = Φ so the complement of a universal set is an empty set.


(i) A ∪ A' = A' ∪ A = ∪ (Complement law)
(ii) (A ∩ A') = Φ (Complement law)
(iii) (A ∪ B)' = A' ∩ B' (De Morgan's law)
(iv) (A ∩ B)' = A' ∪ B' (De Morgan's law)
(v) (A')' = A (Law of complementation)
(vi) Φ' = ∪ (Law of empty set )
(vii) ∪' = Φ and universal set)


Cardinal Number Of A Set
The number of distinct elements in a finite set is called its cardinal number. It is denoted as n(A) and read as 'the number of elements of the set'.


If A and B are finite sets, then
n(A ∪ B) = n(A) + n(B) - n(A ∩ B)
If A ∩ B = Φ , then n(A ∪ B) = n(A) + n(B)
It is also clear from the Venn diagram that
n(A - B) = n(A) - n(A ∩ B)
n(B - A) = n(B) - n(A ∩ B)

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