Set Theory

 

Set Theory

We generally come across different types of collection or groups such as a group of colors, a group of food items, a collection of garments, a collection of books, etc.

This collection or group of objects when mathematically represented is called a set.

Sets are often represented in curly brackets {} and items of the set are called elements or entities.

The German mathematician and logician Georg Cantor created a ‘Theory of sets’ or ‘Set Theory’.

''Set theory is a branch of mathematics that deals with the collection of objects termed as sets.''

Sets should be represented by some rules they are :

• Sets should be symbolized by capital alphabets like A, B, C, D, ……
• Members of the set should be denoted by lower case alphabets like a,b,c,d, …..
• The members belonging to the set is represented by the symbol ‘ε’ called as epsilon and read as “belongs to” for example if a is member of set B it is represented as a ε B and not belongs to is represented as

Set Representation

Sets can be represented in two ways -

1. Roster Form ( Tabular form)     
2. Set builder form

Roster Form ( Tabular form) 

In roster form, the elements of the set are listed in curly braces and are separated by commas. This form is also called the list notation. For example If A is a set containing the list of vowels. It will be represented as A = { a, e, i, o, u}. The order of the elements does not matter. It can also be represented as A = { e, i, a, 0 , u}.

Set Builder Form

In Set builder form all the elements have a common property. This property is not applicable to the objects that do not belong to the set. E.g. If set S has all the elements which are natural numbers less than 8, it is represented as -

S = { x | x is a natural number and x < 10 }  

And it is read as  “the set of all x such that x is a natural number and is less than 10”  in place of x any alphabet can be used.

The roster form of this set would be S = { 1, 2, 3, 4, 5, 6, 7, 8, 9}

Types of Sets : 

Set theory is classified into different types of sets they are -

• Finite set
• Infinite set
• Empty set
• Singleton set
• Equal set
• Equivalent set
• Power set
• Universal set
• Subset

Finite set : Finite sets are the sets having a finite/countable number of members. Finite sets are also known as countable sets as they can be counted. Ex - P = { 0, 3, 6, 9, …, 99}

Infinite set : If any set is endless from start or end or both sides having continuity then we can say that set is infinite. Ex- W = {0, 1, 2, 3, ……..} 

Empty set : The null set, also called the empty set, is the set that does not contain anything. It is symbolized or { }. There is only one null set.  A = {x : 3 < x < 4, x is a whole number} and this set A is the empty set, since there is no whole number between 3 and 4.

Singleton set : A set having exactly one element. Ex- {a}

Equal set : Two or more sets are said to be equal sets if they have the same elements and the same number of elements. For example set A = {1, 2, 3, 4, 5} and B = {1, 2, 3, 4, 5}. Then sets A and B are said to be equal sets as their elements are the same and they have the same cardinality.

Equivalent set : Two sets A and B are said to be equivalent if they have the same cardinality i.e. 

n(A) = n(B).

In general, we can say, two sets are equivalent to each other if the number of elements in both the sets is equal. And it is not necessary that they have same elements, or they are a subset of each other. 

Power set : A power set is set of all subsets, empty set and the original set itself. Ex - Power set of A = {1, 2} is P(A) = {{}, {1}, {2}, {1, 2}}.

Universal set : A universal set is a set which contains all the elements or objects of other sets, including its own elements. It is usually denoted by the symbol 'U'. Ex - A = {2, 4, 6, 8, 10, …} 

Subset : Set A is said to be a subset of Set B if all the elements of Set A are also present in Set B. If set A has {X, Y} and set B has {X, Y, Z}, then A is the subset of B because elements of A are also present in set B.