## Sets

Subject: Compulsory Maths

#### Overview

A set is defined as a collection of well-defined objects, things, numbers, called the elements of the set. The objectives,things or a number of a set are called elements or members of the set. This note contains information about members of sets, a method of describing set, types of the set, set relations, subsets, universal set,set operations Venn diagram.
##### Sets

A well-defined collection of things (objects or numbers, etc) is called set. For examples: 'prime numbers less than 10'. It defines a noticeably

different object which is to be included in the collection. So, 1, 2, 3, 5, and 7 comes under the collection.

Member of a set

The objects belonging to the set are called the members or elements of the set. The membership of a member of a set is denoted by 'belong to' symbol or sign (i.e '$\in$').
For example:
Let's takes set, A = {3, 6, 9, 12, 15}.
In the set A: 6, 9, 12 and 15 are the members or elements of set A. So, '3 $\in$ A' which is read as '3 belongs to set A' or '3 is a member of set A'. Whereas, the number except 3, 6, 9, 12 and 15 does not belong to set A.
(Note: The symbol '$\in$' is used when any elements is not a member of any given set).

Methods of Describing a Set

Generally, set is described by the following three methods:

• Description method
• Listing method
• Set-builder method
1. Description Method:
In this method, the set is described by the common property of a member of set in a sentence inside the braces. For example;
A = {odd numbers less than 10}
E = {even numbers between 7 and 20}

2. Listing Method:
In this method, the set is represented by writing/including its elements inside the braces. For examples:
A = {1, 3, 5, 7, 9}
E = {8, 10, 12, 14, 16, 18}

3. Set-builderMethod:
In this method, the members of a set are represented by a variable like x, y, z, etc. and the variable describes the unique/common property shared by all members (elements) of a set. For examples:
A = {x : x is an odd number less than 10} where is read as A is the set of all values of x, such tat x is an odd number less than 10.
Similarly,
E = { z : z is an even number between 7 and 20}.

Cardinal Number of a Set

The number of elements in a finite set is called the cardinal number of a set. It is denoted by n(A), n(B) n(C) etc. For examples:
A = {2, 4, 6, 8, 10}
Here, the number of elements of sets A is 5. So, the cardinal number of set A is, n(A) = 5
Similarly, M = {5, 10, 15}
∴ The cardinal number of a set M is n(M) = 3.

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. It is denoted by the symbol '⊆'as A ⊆ B. For examples:
A = {whole number less than 6}
i.e. A = {0, 1,2, 3, 4, 5}

B = {odd numbers less than 10}
i.e. B = {1, 3, 5, 7}

C = {even numbers less than 9}
i.e. C = {2, 4, 6, 8}

D = {prime numbers between 1 and 8}
i.e. D = {1, 2, 3 5, 7}

Here, every element of the sets B, C and D is also an element of set A. So B, set C and set D are the subsets of set A.
(Note:An empty set (Φ) is a subset of every set. Every set is a subset of itself)

1. Superset
If a set is a subset of set A, then set A is said to be a superset of B.If is denoted as A⊇ B.

2. Proper Subset
A set 'A' is said to be a proper subset of the set B if it contains at least one element less than set B. It is denoted by the symbol 'C'. Symbolically, we write ABC for A is a proper subset of B. For example:
A = {whole number less than 6}
i.e. A = {0, 1, 2, 3, 4, 5, 6}
B = {odd number less than 7}
i.e. B = {1, 3, 5}
Here, B is a subset of B and the set B is not equal to set A. So, B is a proper subset of B i.e. A⊂ B.
(Note: No set is a proper subset of itself. Null set or empty set is a proper subset of every set).

3. Number of Subsets of a given set
The number of subsets of a given set can be obtained by using the formula '2n', where x is the cardinal number of the given set. For example:
Set A = {a, b}
i.e. n(A) = 2
Here the possible subsets of set A are {a}, {b}, {a, b} andΦ. So, it has 4 subsets.
i.e. 2n = 22 = 4

Universal Set

The universal set is a that has all the elements of other given sets. It is denoted by the symbol U or ξ (pxi). For example:
U = {a, b, c, d, e, f, g, h, i, o,u}
A = {a, b, c, d, e}
B = {a, e, i, o, u}
C = {b, d, f, g, i}
Here, set U is a universal set which is the set of alphabets from a to j and A, B and C are the subsets of universal set 'U'.

Venn - diagram

The diagrammatic representation of sets is called Venn-diagram. It was developed by the British Mathematician John Venn. The universal set

'U' is usually represented by a rectangle and another set is represented by a circle.

Symbols and their Meaning

 $\in$ 'an element of' or 'belongs to' or 'is a member of content' ∉ 'not an element of' or 'does not belong to' or 'is not a member of' ⇒ implies that Iff If and If only /or such that
##### Things to remember
• A collection of well-defined objects is called set.
• Listing method, Description method, and Set builder method are the methods f representing a set.
• Finite or infinite set, null or empty set, singleton or unit set, and universal set are the types of set.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.
##### What is a Subset?

Solution:

Here,
H = {h, e, a, d, s}
T = {t, a, i, l, s}
∴ H∩T = {h, e, a, d, s} ∩{t, a, i, l, s}
= {a, s}

In venn diagram,

Hence, the shaded region represents H∩T.

Solution:

(P∩R)⊂Q
= ({a, b, c, d} ∩ {h, i, e, b}) ⊂ {e, b, f}
= {b} ⊂ {e, b, f} is true (T).
In Venn diagram,

Here, b only is in shaded region i.e. P∩R = {b}
∴ (P∩R)⊂Q is true (T).

(Q∩R)⊂P
= ({e, b, f} ∩ {h, i, e, b}) ⊂ {a, b, c, d}
= {e, b} ⊂ {a, b, c, d} is false (F)
In Venn diagram

Here, the shaded region is Q∩R = {b, e}
∴ Q∩R⊂P is fasle (F).

(P∩Q)⊂R
= ({a, b, c, d} ∩ {e, b, f}) ⊂ {h, i, e, b} is true
∴ P∩Q⊂ R is true (T)
In venn diagram,

Solution:

Here, the relation between M and D, M∩D = Φ
Relation between K and M; M ⊂ K
Relation between K and D; D ⊂ K
Relation between K, M, and D; M∪D = K

D and M are subsets of K and K is subsets of U
K, M and D all are subsets of U.
Now,
the above relation in Venn diagram as is shown below,

Solution:

Here,
From given Venn diagram,
U = {t, u, v, w, x, y, z}
A = {w, x, y, z}
B = {y, u}

Now,
A = {w, x, y, z}, the number of elements in A is 4
∴ The cardinal number of A, i.e. n(A) = 4
Again,
B = {y, u}, the number of element in set B is 2
∴ The cardinal number of B, i.e. n(B) = 2
Then,
A∩B = {w, x, y, z} ∩ {B}
A∩B = {y}
∴ n(A∩B) = 1
And,
We know that,
or, n(A∪B) = n(A) + n(B) - n(A∩B)
or, n(A∪B) = 4 + 2 - 1
∴ n(A∪B) = 5

Similarly,
U = {t, u, v, w, x, y, z}
∴ n(U) = 7

Similarly,
or, ($\overline{A}$) = U - A
or, ($\overline{A}$) = {t, u, v, w, x, y, z} - {w, x, y, z}
or, ($\overline{A}$) = {t, u, v}
∴ n($\overline{A}$) = 3

Similarly,
or, ($\overline{B}$) = U - B
or, ($\overline{B}$) = {t, u, v, w, x, y, z} - {y, u}
or, ($\overline{B}$) = {t, v, w, x, z}
∴ n($\overline{B}$)

Solution:

Here,
P = {a, e, i, o, u}
Q = {a, b, c, d, e}
Now,
Q - P = {b, c, d}
The shaded region represents the elements of Q - P.

P - Q = {i, o, u}
The shaded region represents the elements of P - Q.

The set is a collection  of fine objects .

There are mainly three types of describing set . They are :

• Listing method.
• Set -builder method.
• Description method .

The number of elements contained by a set is known as its cardinal number .

There are mainly three types of sets . They are

• Unit or singleton set.
• Empty or null set.
• Finite and infinite set.

A set is said to be a finite set if it contains a finite number of elements whereas if the sets contain infinite number of elements it is called infinite sets .

If a sets it doesn't contain any elements then they are said to be null sets or empty sets.

The member of sets which are represented by the different variable like x,y,z etc. and the common property of the members is described by the variable is known as set-builder method.