Sets, Relations, & Functions mathematics

Sets, Relations, and Functions are the parts of mathematics that are used in carrying out several logical and mathematical operations. Set is a group of similar types of objects, i.e., the elements. There are different types of sets. The comma “,” separates the elements of a set.

Relations and functions are the set operations that help us connect and work with sets. A set can have many elements, such as a set of chairs, a set of students, a set of cups, etc. You may represent the set of your favourite colours as {yellow, pink, black}.

Set 

In a simple statement, the set is the grouping of similar objects to distinguish them from others. Assume a set named A. If an object ‘x’ is the element of set A, then we shall write it as x ∈A. This means element x belongs to set A. Assume another object y. If the object y is not an element of set A, then we shall write it as y ∉ A. It says that element y does not belong to set A. 

Methods of Representation of Sets

A set has three distinct ways of representations. These are as follows:

• Description form: In the description method, we make a well-defined description of the elements of the set. The description shall be present enclosed in the curly brackets. 

For example: The set of whole numbers less than 24 is as:

{whole number less than 24}

• Roaster form (tabular method): In roaster form or tabular form, we have to list all the elements of the set within curly brackets. The elements have commas as the separation between them.

For example: A set of whole numbers less than 9, in roaster form, is as follows:

A= {0, 1, 2, 3, 4, 5, 6, 7, 8}

As the statement says less than 7, therefore, 7 is not included in it.

• Set Builder form: Set builder form is quite different from the above. In this form, we let any variable (say x) represent any element of the set followed by a certain property. This property must satisfy each element of the set. They are also present inside the curly brackets. 

For example: The set of prime numbers is less than 25.

{x: x is a prime number and x<25}

Types of Sets

A set can be of a variety of types, depending upon its elements. The types of sets are as follows:

• Empty Set: An empty has no elements in it. You may also refer to them as null or void sets. 

For example: {Months of the year having less than 15 days}

There are no months with less than 15 days. Therefore, the set shall remain empty.

• Equal Set: As the name suggests, any two sets with the same number of elements and the elements are same.

For Example: Set A= {a, b, c, d} and set B= {b, a, d, c}, then both sets are equal. 

• Finite Set: A finite set contains a limited or definite number of different elements. In other words, a set with countable elements is a finite set.

For example: Set A= {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}

• Infinite Set: A set that contains an unlimited or uncountable number of elements, that set is the infinite set. 

For Example: A= {set of all-natural numbers}

• Singleton Set: A set that contains only a single (one) element is the singleton set or unit set.

For example: {x: x is capital of India}

Subsets: Let A and B be two sets. If every member of B is also a member of set A, then B is the subset of A.

For Example: If A= {1,2,3,4,5} and B={4,2,3}, then every member of set B is a member of set A. Thus, we can say that B ⊂ A. 

Relation and Functions

 A relation in mathematics denoted by R is the subset of the Cartesian product. It contains a few of the ordered pairs. These pairs have a relationship defined between the first and second elements of the set. 

In case every element of a particular set, say A is related with only a single element of another set. This type of relationship is defined as the function. We can also say that A function is a unique case of relation with no two ordered pairs with a similar initial element.

Types of Functions

• One-to-one function
• Onto function
• Constant function
• Bijective Function
• Algebraic function
• Identify function

Conclusion 

Sets, Relations, and Functions are the parts of mathematics, i.e., used in carrying out several logical and mathematical operations. Set is a group of similar types of objects, i.e., the elements. There are different types of sets. The comma “,” separates the elements of a set.

Relations and functions are the set operations that help us connect and work with sets. A relation in mathematics denoted by R is the subset of the Cartesian product. A function is a unique case of relation with no two ordered pairs with a similar initial element. There are different types of functions. Sets, relations, and functions are an important part of algebra and mathematics.