What is a Subset?
- A collection of elements is known as a subset of all the elements of the set contained inside another set
- Example:
- Set A has {X, Y}
- Set B has {X, Y, Z},
- Then A is the subset of B because elements of A are also present in set B
Subset Symbol
In set theory, a subset is denoted by the symbol ⊆ and read as ‘is a subset of’.
A ⊆ B
Subsets of set A = {1, 2, 3 , 4}
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- {}
- {1}, {2}, {3}, {4},
- {1,2}, {1,3}, {1,4}, {2,3},{2,4}, {3,4},
- {1,2,3}, {2,3,4}, {1,3,4}, {1,2,4}
- {1, 2, 3, 4}
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Types of Subsets
- Proper Subset
- Improper Subsets
Proper Subset
A proper subset is one that contains few elements of the original
- Set B is considered to be a proper subset of Set A if Set A contains at least one element that is not present in Set B.
- Example set A : {2, 4, 6}, then,
- Number of subsets : {2}, {4}, {6}, {2,4}, {4,6}, {2,6}, {2,4,6} and Φ or {}.
- Proper Subsets : {}, {2}, {4}, {6}, {2,4}, {4,6}, {2,6}
- Proper Subset Symbol
- A proper subset is denoted by ⊂ and is read as ‘is a proper subset of’. Using this symbol,
- B: {12, 24} and A: {12, 24, 36},
- B ⊂ A
- The formula to calculate the number of proper subsets is 2n – 1
- where n is the number of elements in the set
- A: {12, 24, 36}, no of proper subsets= 23 – 1 = 7
- Proper Subsets : {}, {12}, {24}, {36}, {12, 24}, {24, 36}, {12, 36}
Improper subset
The improper subset contains every element of the original set along with the null set.
- For A: {12, 24, 36} All subsets : {}, {12}, {24}, {36}, {12, 24}, {24, 36}, {12, 36}, {12, 24, 36}
- The empty set is an improper subset of itself
Power set
A power set is a set that includes all the subsets including the empty set and the original set itself. It is also a type of set.
- If set A = {x,y,z}
- all subsets {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z} and {}
- Power set of A, P(A) = {x}, {y}, {z}, {x,y}, {y,z}, {x,z}, {x,y,z} and {}
- Where P(A) denotes the powerset.
- The number of elements of a power set is written as |A|,
- If A has n elements then it can be written as:
- |P(A)| = 2n
Example of Power set
Q.1: Find the power set of Z = {2,7,9} and the total number of elements.
Solution:
Given, Z = {2,7,9}
Total number of elements in powerset = 2n
Here, n = 3 (number of elements in set Z)
So, 23 = 8, which shows that there are eight elements of power set of Z
Therefore,
P(Z) = {{}, {2}, {7}, {9}, {2,7}, {7,9}, {2,9}, {2,7,9}}
Q.2: How many elements are there for the power set of an empty set?
Solution:
An empty set has zero elements.
Therefore, no. of elements of powerset = 20 = 1
Hence, there is only one element of the powerset which is the empty set itself.
P(E) = {}
Cardinality of powerset
- The cardinality of the power set is the number of elements present in it.
- It is calculated by 2n where n is the number of elements of the original set.
Properties of subset
- Every set is considered as a subset of the given set itself. It means that X ⊂ X or Y ⊂ Y, etc
- We can say, an empty set is considered a subset of every set.
- X is a subset of Y. It means that X is contained in Y
- If a set X is a subset of set Y, we can say that Y is a superset of X
Constructing Subset
A subset of Even Integers
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- {X l X Z, X mod 2=0}
Set of a perfect square
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- {X l X N, √X N}
Set of Rational numbers in reduced form
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- { l p , q Z, gcd( p , q)=1}
Set of integers from -6 to 6
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- {X l X Z,-6 x 6}
Set of real numbers between 0 & 1
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- {X l X R, 0 < x < 1}
Set of real numbers between 0 & 1(including 1 ) (0,1]
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- {X l X R, 0 < x 1}
Set of real numbers between 0 & 1(including 0 ) [0,1)
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- {X l X R, 0 x < 1}
Divisibility Pairs of natural numbers
- (d, n) such that d|n
- Pairs such as (7, 63), (17, 85), (3, 9) . . .
- D = {(d, n) | (d, n) ∈ N × N , d|n}
Divisibility Pairs of Integers
- E = {(d, n) | (d, n) ∈ Z × Z, d|n}
- Now (−7, 63), (−17, 85), (−3, 9) . . .
Prime powers Pairs of natural numbers (p, n) such that p is prime and n = p m for some natural number m
- get prime no: P= { p | p ∈ Z , factor (p)=(1,p) & p 1}
- Y= {(p, n) | (p,n) ∈ N x N, n = p m . & m ∈ N}