Superset, Power Set, Universal Set

All Subsets of a Set

The subsets of any set consists of all possible sets including its elements and the null set.

 Let us understand with the help of an example.

Example: Find all the subsets of set A = {1,2,3,4}

Solution: Given, A = {1,2,3,4}

Subsets =    {}

{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}

Power Set

The power set is the set of all subsets that can be created from a given set. The cardinality of
the power set is 2 to the power of the given set’s cardinality.
Notation: P (set name)
Example: A = {a, b, c} where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, {a, b, c},{} } and |P (A)| = 8
Note: In general, if |A| = n, then |P (A) | = 2n

Superset

If A ⊂ B and A ≠ B, this means that A is a proper subset of B. And is known as the superset of set A. For e.g. all natural numbers are integers. If N and Z represent the set of all the natural numbers and integers, respectively, then we can write that N ⊂ Z

Here, N is a proper subset of Z and Z is called the superset of N

Universal Set

A set which contains all the sets relevant to a certain condition is called the universal set. It is the set of all possible values. 

Example: If A = {1,2,3} and B {2,3,4,5}, then universal set here will be:

U = {1,2,3,4,5}

Let P be the set of prime numbers and let S = {t | 2t – 1 is a prime}.
Prove that S ⊂ P.
Solution Now the equivalent contrapositive statement of x ∈ S ⇒ x ∈ P is x ∉ P ⇒
x ∉ S.

 

Now, we will prove the above contrapositive statement by contradiction method
Let x ∉ P
⇒ x is a composite number
Let us now assume that x ∈ S
⇒ 2x – 1 = m (where m is a prime number)
⇒ 2x = m + 1
Which is not true for all composite number, say for x = 4 because
24 = 16 which can not be equal to the sum of any prime number m and 1.
Thus, we arrive at a contradiction
⇒ x ∉ S.
Thus, when x ∉ P, we arrive at x ∉ S
So S ⊂ P.