Sets, Relations and Functions

Sets, relations, and functions help conduct logical and mathematical set operations on a mathematical and real-life basis. Sets help in differentiating the group of certain kinds of objects. Sets can be represented in three ways: statement form, roaster form, also known as roaster form, and set builder form. To learn more about this topic, keep reading our study material notes on sets, relations, and functions. 

What are Sets?

Sets are a collection of well-defined objects. These objects have one or multiple common properties. By grouping the objects, we distinguish objects from members of different sets. If A is a set and “a” is an element, then an ∈A means that ‘a’ belongs to set ‘A’. Sets are represented in three ways:

• Statement Form: Here, the one statement describes all the elements inside the set. For example, V= all vowels in English. 

• Roaster Form: In this form, every member in the given set is written in a pair of brackets. Commas separate them. For example: E= { 2, 4, 6, 8, 10, 12}. This is a set of even numbers between 1 to 12. 

• Set Builder Form: Here, the stated property should be common in all the elements of that set. For example A = {x l x, all even natural number under 15} then A = { 2, 4, 6, 8, 10, 12, 14}.

Types of Sets 

There are nine different types of sets. They are as follows:

1. Empty Sets 

A set that has no elements. These are also known as null or void sets. It is denoted as A={} or A = ∅

2. Singleton Set 

In this set, there is only one element. For example, A={5}

3. Proper Set 

In this type, if there are two sets, then A is a proper subset of B. for example, B=( 2, 3, 5} and A={ 2, 5} then A is a proper subset of B. 

4. Power Set

Power sets consist of all the subsets of a given set. A power set is denoted by P(A); here, the number of elements in the set has some power. For example, if A is { 1, 2 } then P(A) is { (∅), (1), (2), (1 , 2)}

So the number of elements in P(A) is 2 ² which is 4. 

5. Finite Set 

There are only a certain number of elements in the set. For example D={ 2, 4, 6, 8, 10}. There are 5 objects in set D. 

6. Infinite Set 

In this set, there are infinite numbers. For example: B={2,4,6,8,10,…………..}

7. Universal Set 

In this type, any set is a superset of all sets under consideration. These sets are either denoted by S or U. Let us consider C={ 3, 4, 7} and D ={ 1, 2,3} then in universal set U={1,2,3,4,7}

8. Equal Sets 

There are two sets, and if they are both equal, then they are subsets of each other. For example, If P ⊆ Q and Q ⊆ P, then they both are equal.

P{3,6,9} and q={6,9,3}.

P and Q have the same elements in the above sets, so they are known as equal sets. 

9. Disjoint Sets 

In these sets, there are no common elements. for example: P={1,2,3} and Q= {4,5,6}

Terms related to Relation and Function 

Relations and functions define a map between the two sets. A relation is a non-empty set and is a subset of the cartesian product. The subset is derived by describing the relation between the elements of the pair. The function is a relation between two sets. Each element of one set has only one image in the other set. There are certain terms which you should be aware of:

• Cartesian Product: There are two non-empty sets A and B, the cartesian product A x B. For example, X = {1, 3} and Y = {4, 7} then, X × Y = {(1, 4), (1, 7), (3, 4), (3, 7)}.For the above example, the number of elements in X is 2 and the number of elements in Y is 2 then the number of ordered pairs in the Cartesian product n(X × Y) = 4. 

• Domain: It is a set of all the first elements of the ordered pair, which are about set A and set B. It is also known as a set of inputs or pre-images. 

• Range: The range is a set of all second elements of the ordered pair, which are about set A and set B. They are known as a set of outputs or images. 

• Codomain is the whole range of the relation from set A and set B. 

Conclusion

Relations and functions are set operations that help identify the relationship between two elements of two or multiple sets or even between the elements of the same set. Based on the kind of relationship the elements have, they are categorised into different functions. I hope this study material notes on sets, relations, and functions can better understand this concept and help people prepare for competitive exams.