Subset

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}

• • • {}
    • {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}

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

• • {X l X Z, X mod 2=0}

Set of a perfect square

• • {X l X N, √X N}

Set of Rational numbers in reduced form

• • { l p , q Z, gcd( p , q)=1}

Set of integers from -6 to 6

• • {X l X Z,-6 x 6}

Set of real numbers between 0 & 1

• • {X l X R, 0 < x < 1}

Set of real numbers between 0 & 1(including 1 ) (0,1]

• • {X l X R, 0 < x 1}

Set of real numbers between 0 & 1(including 0 ) [0,1)

• • {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}