Subject: Compulsory Maths
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 |
Represent the sets H = {h, e, a, d, s} and T = {t, a, i, l, s} in a Venn-diagram and write H∩T by listing method.
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.
If P = {a, b, c, d}, Q = {e, b, f} and R = {h, i, e, b}; state which of the following is true or fasle.
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,
Draw a Venn diagram to illustrate the relation among the given sets:
U = {Students of Nepal}
K = {Students of Kathmandu valley}
M = {Students of morning shift of Kathmandu}
D = {Students of day shift of Kathmandu}
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,
Look at the given Venn diagram and find the cardinal number of the following sets:
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(\(\overline{B}\))
If P = {a, e, i, o, u} and Q = {a, b, c, d, e} find Q - P and P - Q..
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.
What is sets ?
The set is a collection of fine objects .
How many methods of describing sets are there and what are they ?
There are mainly three types of describing set . They are :
Define Cardinal number of sets ?
The number of elements contained by a set is known as its cardinal number .
How many types of sets are there ? Write down its name also ?
There are mainly three types of sets . They are
What is finite and infinite sets ?
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 .
What is null sets ?
If a sets it doesn't contain any elements then they are said to be null sets or empty sets.
What is set -builder method ?
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.
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