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:
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)
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 |
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:
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,
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.
Which symbol is used to indicate the Equivalent sets?
Universal set is denoted by ______ .
The sets which have all the elements of other given sets, that is known as ______.
An empty set is denoted by ______.
An empty set is also known as ______.
If the sets have at least one common elements then it is said to be _______ sets.
If A = {2, 4, 6, 8, 10} and B = {1, 2, 3, 4, 5, 6, 7}, then find A ∪ B.
A = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} is a ______ set.
The number of elements in a finite set is called ______ of a set
How many types of sets are there ?
ASK ANY QUESTION ON Sets
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