Subset, Proper Subset and Power Set

Subset Definition

A set A is said to be a subset of a set B if every element of A is also an element of B. To denote A is a subset of B the subset symbol ⊂ is used. We can write it symbolically as A ⊂ B. The subset symbol ⊂ stands for ‘is a subset of’ or ‘is contained in’.

In other words A ⊂ B if whenever a ∈ A, then a ∈ B. Thus

A ⊂ B if a ∈ A ⇒ a ∈ B

Where the symbol ‘⇒’ means implies

If A is not a subset of B we write A ∈ B

From the definition of subset we can also say that every set A is a subset of itself, i.e. , A ⊂ A. As the empty set ∅ has no elements we say that ∅ is a subset of every set.

If A ⊂ B and B ⊂ A then A = B or if two sets are subsets of each other than the two sets are equal sets.

Proper Subset

If A ⊂ B and A ≠B then A is called proper subset of B and B is called superset of A.

Example of proper subset can be A = { 4, 5, 6} is a proper set of B = { 4, 5, 6, 7, 8}.

Singleton Set

The set having only one element is called singleton set. Thus {b} is a singleton set.

Some Subset of R

The set of natural numbers, the set of whole numbers, the set of integers, the set of rational numbers, the set of irrational numbers all are R subset.

The set of all natural numbers is denoted as N; where N = {1, 2, 3, … }

The set of all whole numbers is denoted as W; where W = {0, 1, 2, 3, … }

The set of all integers numbers is denoted as Z where Z = {…, -3, -2, -1, 0, 1, 2, 3, … }

The set of all rational numbers is denoted as Q where Q= {p/q ∶p,q ∈Z,q ≠0}

The set of all irrational numbers is denoted by Q’ where Q^'= {x∶x∈R and x ∈ Q} where R denotes the set of real numbers. √2, √3, π, e are all elements of Q’

N, W, Z, Q, Q’ are all subsets of R, where R is the set of real numbers. It is obvious that

N ⊂ W ⊂ Z ⊂ Q ⊂ R

Q’ ⊂ R & N ⊂ Q’

Interval Notation

Let a, b εR and a < b . Then the set of all real numbers between a and b is denoted in interval notation form as (a,b) and is called an open interval.

Thus (a,b) = { x εR : a < x < b) .

This (a,b) is the interval of all real numbers between a and b, excluding both a and b.

All the points between a and b belong to the open interval (a,b) but a and b do not belong to this interval.

The interval which contains the end points is also called closed interval and is denoted by [a,b]. Thus

[a,b] = { x εR : a ≤ x ≤ b }

This [a,b] is the interval of all real numbers between a and b, including both a and b.

We can also have intervals closed at one end and open at the other.

[a,b) = { x εR : a ≤ x < b}

This [a,b) is the interval of all real numbers between a and b, including a but excluding b.

(a,b] = { x εR : a < x ≤ b}

This (a,b] is the interval of all real numbers between a and b, excluding a but including b.

Therefore in interval notation the set of positive real numbers R+ can be written as

R+ = (0,∞)

In interval notation, the set of negative real numbers is written as (-∞ ,0)

The set R itself in interval notation is given as (-∞ ,∞)

The length of any interval (a,b) or [a,b] or [a,b) or (a,b] is given by b – a

Power Set

The collection of all subsets of A is called the power set of A.

It is denoted by P(A). Every element of P(A) is a set.

The power set of any set always contains the null set and the set itself.

Example of Power Set of a Set

If A = { 3, 4} then the power set of set A is written as P(A) = {∅,{3},{4},{3,4}}

If set A has n elements then the power set of the set A, P(A), has 2n elements.

R is said to be a universal set because N, W, Z, Q, Q’ are subset of R.