Sets and Types of Sets

A set is a collection of distinct objects(elements) which have common property. For example, cat, elephant, tiger, and rabbit are animals. When, these animals are considered collectively, it's called set.

Set Notation

The members(elements) of set is separated by comma and braces { } are used outside the comma separated elements.

{cat, elephant, tiger, rabbit}

For convenience, sets are denoted by a capital letter. For example,

A = {cat, elephant, tiger, rabbit}

Here, A is a set containing 4 elements.  

Important Points Regarding Sets

  1. A Set is collection of distinct members.
    		Wrong: Elements are not distinct
               A = {a, b, c, b, c, d}
    
    		Right: Elements are distinct
    		   A = {a, b, c, d}
    
  2. The number of elements in a set can be both finite and infinite.
              N = {2, 4, 6}
              N = {..., -2, -1, 0, 1, 2, ...}
    
  3. The elements of a set can be in any order.Changing the order of elements doesn't change anything.
             A = {1, 2, 3}
    
             This set can also be written as:
             A = {2, 1, 3}
             A = {3, 1, 2} and so on 

Example of Sets

Example #1: What is the set of all vowels in English alphabet?

V  = {a, e, i, o, u}

Example #2: What is the set of integers between 2 and 9?

I = {3, 4, 5, 6, 7, 8}

Example #2: What is the set of prime number?

P = {2, 3, 5, 7, 11, ... }

Types of Sets

Empty set

A set which do not have any element is known as empty set. It is also called Null Set, Vacuous Set or Void Set. Empty set is denoted by ϕ.

A = {} =ϕ

Singleton set

If a set has only one element, it's known as singleton set. 

A = { moon }

Finite set

Set with finite number of elements is called finite set.

S = {1, 2, 3}

​Infinite set

A set with have infinite number number of elements is called infinite set.

A= { x:x is an integer }
B = { 5, 10, 15, 20, 25, ... }

Equivalent sets

Two sets are said to be equivalent sets if they have same number of elements. For Example,

A = {a, b, c, d}
B = {e, f, g, h}

Here, A and B are equivalent sets because both sets have 4 elements.

Equal Sets

Two sets are said to be equal sets if they both have exactly same elements.

A = {1, 2, 3, 4}
B = {2, 4, 3, 1}

Here, A and B are equal sets because both set have same elements (order of elements doesn't matter).

Overlapping Sets

Two sets are said to be overlapping sets if they have at least one element common.

A = {1, 2, 3, 4}
B = {3, 4, 5}

Here A and B are overlapping sets because elements 3 and 4 are common in both sets.

Disjoint Sets

Two sets are said to be disjoint sets if they don't have common element/s.

A = {1, 2, 3, 4}
B = {5, 6}

Here A and B are disjoint sets because these two sets don't have common element.

Subset

A set P is a subset of set Q if every element of set is also the member of set Q.  Simply, if set P is contained in set Q, P is called subset of superset Q. It is denoted by P⊂Q.

P = {1, 2, 3}
Q = {1, 2, 4, 3, 9}

Here, all three elements 1, 2, and 3 of set P is also member of set Q. Hence, P is subset of Q.