Pre- Requisites of Sets

A set is a well-defined group of items whose elements are constant and cannot change. It signifies that the set does not change from one individual to the next. The set of natural numbers up to 7 will, for example, remain the same as 1,2,3,4,5,6,7. Still, if we say the best players on a football team, the names of the players may differ each time we ask about the best players, because each person has their own opinion of who the best player is. Similarly, if we speak about the set of rivers in India, the constituents of the set will remain the same. So, here’s a real-life set example. Curly brackets are used to represent sets in mathematics.

Origins of Set Theory

Georg Cantor (1845-1918), a German mathematician, was the first to propose the concept of ‘Set Theory.’ He came upon sets while studying on “Problems on Trigonometric Series,” which has since become one of the most fundamental notions in mathematics. Concepts such as relations, functions, sequences, probability, geometry, and so on was impossible to explain without first comprehending sets.

Definition of sets:

A set is a well-defined collection of objects and entities.

Representation of Sets

There are two ways in which we can  express a set:

Roaster form

Set Builder Form

Roster Form

All of the set’s elements are listed in roster form, separated by commas and encased in curly braces.

For instance, if set represents all leap years between 1995 and 2015, it would be written in Roster form as:

A ={1996,2000,2004,2008,2012}; B ={1996,2000,2004,2008,2012}; C ={1996,2000,2004,2008}

Inside the braces, the components are now written in ascending sequence. This could be in any order, including descending or random. 

Set builder form

All of the elements in the set builder form share a common property. This property does not apply to items that are not part of the set.For instance, if set S contains all even prime numbers, it is expressed as:

S={x: x is a prime number that is even} where ‘x’ is a symbolic representation of the element that is used to describe it. ‘:’ stands for ‘in such a way that’. 

As a result, S = {x:x is an even prime number} means “the set of all x such that x is an even prime number.” S = 2 would be the roster form for this set S. 

Types of sets

Empty Sets

An empty set is a set that contains no or null entries. This is also known as a Void set or a Null set. The symbol {} is used to denote empty set

Consider the following scenario: Set X ={ x:x} represents the number of pupils in Class 6th and Class 7th.

Set X is an empty set because we know a student cannot learn in two classes at the same time.

Singleton set

A singleton set is defined as a set with only one element.

Set X ={ 2} is a singleton set, for example.

Finite and Infinite Sets

Finite sets are those with a finite number of members, whereas infinite sets are those with an unknown number of elements but a figure or number that is too vast to describe in a set.

Set X = {1, 2, 3, 4, 5} is a finite set since it has a finite number of items.

Because there is an approximate number of Animals in India, set Y = Number of Animals in India is an infinite set. However, the actual value cannot be expressed because the numbers could be very large.

Equal Sets

If every element of set X is also an element of set Y, and if every element of set Y is also an element of set X, two sets X and Y are said to be equivalent. It signifies that the elements in sets X and Y are the same, and we can denote this ;Y = X

Let’s say X = {1, 2, 3, 4} and Y ={ 4, 3, 2, 1} then X = Y.

And if X = set of even numbers and Y = set of natural numbers, then X = Y, because natural numbers include all positive integers from 1 to infinity, but even numbers begin with 2, 4, 6, 8, and so on.

Subsets

If the elements of a set X belong to set Y, it is said to be a subset of set Y, or each element of set X is present in set Y. It is represented by the symbol X Y.

The subset notation can also be written as;

If an X exists, then a Y exists.

ay

“X is a subset of Y if an is an element of X implies that an is likewise an element of Y,” says the above equation.

A null set or empty set is a subset of all sets, and each set is a subset of its own set.

Power Sets

 The collection of all subsets is called power set. Let us show you how.

Every set is called a subset of itself, and the empty set is a subset of all sets. Consider the set X = 2, 3 as an example. We can deduce the following from the preceding statements:

{}is a subset of 2 and 3.

{2} is a subset of 2 and 3.

{3} is a subset of 2 and 3.

{2, 3} is a subset of 2, 3 as well.

As a result, the power set X = {2, 3}

P(X) ={,2,3,2,3,3,3}

Universal Sets

A universal set is a set that includes all items from other sets. The letter ‘U’ is commonly used to represent it.

Set X = {1, 2, 3} and Y = {3, 4, 5, 6}  as an example.

Then we can write U = 1, 2, 3, 4, 5, 6, 7, 8, 9, for a set.

Note: According to the universal set’s definition, all sets are subsets of the universal set. Therefore,

X  U

Y⊂ U

Union of sets

The elements of two sets are combined in a union. It is denoted by the symbol.

Set X = {2, 3, 7} and Y = {4, 5, 8} as an example.

Then set X and set Y will be joined;X U Y = {2, 3, 7, 4, 5, 8}, 

Union of Sets Properties:

1. Commutative law: X UY = YU X.
2. (XUY)UZ = Z U(XUY) 
3. X U X=X
4. U U X=X

Intersection of Sets

The intersection of sets is the set of all elements that are common to all of the specified sets. It’s represented by the symbol ∩

Set X = {2, 3, 7} and Y = {2, 4, 9} as an example.

As a result, X ∩ Y = 2

Difference of Sets

The difference between sets X and Y is that it only contains components that belong to set X and not to set Y. i.e. X – Y = {a}: there is an X and a Y.

If X = {a, b, c, d} and Y = {b, c, e, f}for example, then

X – Y = {a, d}; Y – X = {e, f}; X – Y = {g, h}; X – Y = {g, h}; X – Y ={ g, h};

Disjoint Sets

Disjoint sets are created when two sets X and Y have no common items and their intersection produces zero(0).

X ∩ Y = 0 is one way to express it.

Conclusion:

This article gives a brief insight about the pre requisites of sets; sets are well defined collection of objects and entities which are represented within curly braces. Further the article showers light upon origin and representation of sets. Also the article talks about various types of sets and operations associated to it.