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Sets

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

source :1080.plus Fig :Introduction to Set Concepts & Venn Diagrams
source :1080.plus
Fig :Introduction to Set Concepts & Venn Diagrams

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

source :finitegeometry.org Fig: Venn Diagrams and Finite Geometry
source :finitegeometry.org
Fig: Venn Diagrams and Finite Geometry

'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



  • 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.
.

Very Short Questions

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,

s

Hence, the shaded region represents H∩T.

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,

s 

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.

0%
  • Which symbol is used to indicate the Equivalent sets?

    A ↔ B
    A ≈ B
    A ≡ B
     A ∼ Q
  • Universal set is denoted by ______ .

    ξ
    µ
    Ω
    ς
  • The sets which have all the elements of other given sets, that is known as ______.

    Universal set
    Equivalent set
    Overlapping set
    Equal set
  • An empty set is denoted by ______.

    ο


    Φ
  • An empty set is also known as ______.

    Cardinal set
    Disjoint set
    Null set
    Singleton set
  • If the sets have at least one common elements then it is said to be _______ sets.

    disjoint
    equivalent
    equal
    overlapping
  • If A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6, 7}, then find A ∪ B.

    A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
    A ∪ B = {1, 3, 5, 7, 9,}
    A ∪ B = {2, 4}
    A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8, 10}
  • A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is a ______ set.

    universal
    equivalent
    unit
    finite 
  • The number of elements in a finite set is called ______ of a set

    relation
    membership 
    method
    the cardinal number
  • How many types of sets are there ?

    5
    2
    4
    3
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