What are Set Notations and how are they used in Maths?

 Introduction

Any collection of objects (elements) in mathematics and logic, which may or may not be mathematical (e.g., numbers and functions). A list of all the members of a set is often expressed using braces. The intuitive concept of a set is likely older than the concept of a number. A herd of animals, for example, could be matched with stones in a sack without either pair being numbered. The concept goes on indefinitely.

Set Notation:

The elements and attributes of sets are defined using symbols in set notation. When writing and explaining sets, symbols help you save space.

Set notation also allows us to use symbols to indicate distinct relationships between two or more sets. We can easily do operations on sets like unions and intersections this way.

Set notation can appear at any time, and it could even be in your algebra lesson! As a result, understanding the symbols used in set theory is advantageous.

Set notation is a set of symbols that can be used to:

• defining elements of a set
• Show links between sets
• Demonstrate operations between sets

Denoting a set:

A set is traditionally represented by a capital letter, whereas its elements are represented by lower-case letters.

Commas are commonly used to separate the parts. For example, the set A that comprises the vowels of the English alphabet can be written as:

     A = {u , o , i , e , a}

We read it as ‘The set A comprising the vowels of the English alphabet,’.

Membership of a set:

To indicate membership in a set, we use the symbol ∈.

B = {1 , 3 , 4 , 5 , 7}

We write 1 ∈ B and read it as ‘1 is an element of set B’ or ‘1 is a member of set B’ because 1 is an element of set B.

We write 6 ∉ B and read it as ‘6 is not an element of set B’ or ‘6 is not a member of set B’ because 6 is not an element of set B.

Specifying Members of a Set:-

Set B can be described as follows using the set-builder notation:

C = {x : x ∈ N ; x ≤ 5}

‘The set of all x such that x is a natural integer less than or equal to 5’ is how we understand this notation.

Subset of a set:-

When every element of set A is also an element of set B, we say that set A is a subset of set B. A can also be said to be enclosed within B. The following is the notation for a subset: A⊆B

‘is a subset of’ or ‘is contained in’ is represented by the symbol. ‘A is a subset of B,’ or ‘A is contained in B,’ is how we generally read A⊆B.

To establish that A is not a subset of B, we use the notation below: A⊈B

Because the sign denotes ‘is not a subset of,’ we read A⊈B as ‘A is not a subset of B.’

Equal Sets:-

We say set A is equal to set B if every element of set A is also an element of set B, and every element of B is also an element of A.

To prove that two sets are equal, we use the notation below.

A=B means “set A equals set B” or “set A is the same as set B.”

The Empty Set:-

A set containing zero elements is called an empty set. It’s also known as a null set. The empty set is represented by the symbol ∅ or by empty curly braces {} .

It’s also worth mentioning that every set has an empty subset.

The Universal Set:-

The universal set is a collection of all the items being considered. The universal set is traditionally represented by the symbol U.

Power Set:-

Set A’s power set is the set that contains all of A’s subsets. A power set is denoted by P(A) and is read as ‘the power set of A.’

The union of two sets:-

The set that contains all elements from both sets A and B is known as the union of the two sets.

The union of A and B is denoted by A U B, which is read as ‘A union B.’ The union of A and B can also be defined using the set-builder notation, as illustrated below.

A U B = {x : x ∈ A or x ∈ B}

All of the items in each of the sets are contained in the union of three or more sets.

If an element belongs to at least one of the sets, it is a member of the union.

The intersection of two sets:-

The intersection of sets A and B is the set that contains all of the elements from both sets.

The intersection of A and B is denoted by A ∩ B and read as ‘A intersection B.’

The intersection of A and B can also be defined using the set-builder notation, as illustrated below:

A ∩ B = {x : x ∈ A and x ∈ B}

Difference of two sets:-

The set difference between sets A and B is the set of all elements in A that are not found in B.

The set difference between A and B is denoted by A\B or A-B, and it is read as ‘A difference B.’

A and B’s set difference is also known as the relative complement of B with respect to A.

In set builder form we can define it as 

A\B = A – B = {x : x ∈ A and x ∉ B}

Conclusion:-

The symbols used in the process of working across and within sets are referred to as set notation. Curly brackets are the most basic set notation for representing set items. A = {a, b, c, d }is one example of a set. The set is signified by a capital letter, and the elements by tiny letters.