Sets, Relations, & Functions

Sets, Relation, and Functions are some of the most significant topics when we talk about set theory. In Mathematics, Sets, Relations, and Functions set up the base and serve as the basic building for the whole calculus. Some students find this topic very confusing, but this would be the easiest topic you would ever come across once you start solving problems. Sets, relations, and functions are three different words with different meanings but equal importance while studying mathematics. Sets, Relations, and Functions have a wide variety of examples in our real lives as well, let us have a look at some of them. 

Examples of Sets, Relations, and Functions

Sets: Let us take an example of your favourite song playlist. Now, when you design a playlist, you add songs to that playlist. Now, your playlist is a SET that contains various songs of your choice. 

Relations: For relations, let us take a situation where the time of the day is increasing, the temperature also increases. Now, at every point in time, we will have a unique temperature related to that time. The relation of a unique time with a unique temperature in the language of mathematics is called Relations. 

Functions: Functions can be understood by an example of a machine that generates electricity. The unique output (electricity) generated for the input we give in any form, in mathematical language, is called a function. 

Students generally get confused between Relations and Functions, thus you need to keep in mind that, “All functions are relations, but all relations are not functions.”

To better understand this topic, let’s dive deeper and learn these three different mathematical expressions in varied forms. 

What are Sets? 

A set can be defined as a collection of distinct and well-defined objects. 

NOTE- The objects under a set should always be distinct 

A collection of sets is preferably known as a family or combination of sets. 

For example- {A1, A2, A3, A4,….. An ) is a family of the set with distinct and well-defined objects. This can be further denoted as S= {Aj where j belongs to N and 1 ≤ j ≤ n} 

Sets are denoted by a capital letter and the objects in the sets are defined inside { } curly braces. 

Types of Sets

There are numerous types of sets in mathematics, out of which some are discussed below:

1- Singleton Set

As the name suggests, these sets contain only a single object. For example, A ={3} or B= {Box}. 

2- Empty Set

An empty set is also known as a Null set. This set has no object inside it. For example, A= { } 

3- Proper Set

If A and B are two sets with distinct objects, A is a proper subset of B if A ⊆ B is but A ≠ B.

For example, A= { 1,2,3 }, B= {1,2,3,4}. Here, A contains objects of B but A ≠ B thus, A is a proper subset of B. 

4- Universal Set

The universal set can be termed as the combination of all the sets to be considered. Generally it is denoted by S or U. For example, if A={1,2,3} and B= {3,4,5} then, S= {1,2,3,4,5}. 

5- Infinite and Finite Sets

As their name suggests, if a set has a finite number of objects, it is considered a Finite set, and if a set has an infinite number of objects, it is considered an Infinite set. 

For example, A= {1,2,3} is a finite set and B={1,2,3…..} is an infinite set. 

What are Relations in Maths?

Relations in set theory are helpful at times when you need to find common relations between the input or output of any functions. They are also used to define a relation between two sets. 

For example- if A= {a,b,c} and B={1,2} then let R= {(a,1), (a,2), (b,1), (b,2), (c,1), (c,2)}

Here R is a proper subset of AxB. 

Types of Relations

There are various types of relations in mathematics, some of which are given below:

1- Empty Relation

An empty relation can be termed as when there is no relation between two sets. 

For example, consider a basket of 100 apples. There are no mangoes in them. Thus, there is no relation between these two sets.

2- Universal Relation

A universal relation can be termed as when every function of a set is related to each function of a set, i.e., R=A x A 

3- Inverse Relation

An inverse relation is observed when A= { (a,b), (c,d) } then the inverse of this relation will be A-1 = { (b,a), (d,c) }. Thus, inverse relation can be defined as the inverse of the same set. 

4- Reflexive Relation

In a reflexive relation, every object of a set maps for itself. 

For example, if a set A={ 1,2 }, then the reflexive of this relation would be R= {(1,1), (2,2), (1,2), (2,1)}  

5- Symmetric Relation

As the name suggests, symmetric relations can be termed as the relations where a=b ∈ R then (b, a) ∈ R. A very good example of symmetric relations can be, R = {(2, 3), (3, 2)} for a set A = {2, 3}. 

What are functions?

Functions can also be seen as a special type of relations as according to functions, there should be only one unique output for all the inputs. Functions are simply used to show the dependency of an object of relation with the other object of relation. 

Types of Functions

Functions are broadly categorised into three categories, which are: 

1- Injective Function

Injective functions or one-to-one functions can be defined as if for each element of a set P, the set Q has a unique and distinct output. 

2- Many to One Function

Many to one functions are the functions where two or more elements of the set P are matched with a single element of the set Q. 

3- Bijective Function 

If a function matches with each element of set P with a discrete element of set Q, then, such types of functions are known as bijective functions. 

Conclusion

Now we know about all the three important pillars of Set theory. With this article, we covered almost all the aspects of sets, functions, and relations. The set theory covers a very major part of Algebra in mathematics. Sets, Relations, and Functions are very conceptual topics where one must be fully focused on what they are studying.