Let’s Learn About Sets

What is a set? Well, simply we can say it’s a collection of things. Sets in mathematics are simply an organized collection of different objects forming a group. A set can be a group of any number of items, it can be a collection of numbers, shapes, symbols, days of a week, types of vehicles, and so on. For example, the items you have in your college bag or school bag: Notebooks, Books, Files, pens, pencils, and so on. I am sure you can also come up with many other items. This is called a set. So it is just items or things grouped together with some common features or properties.

Definition of Sets

Set is any collection of objects which may be mathematical or not and whose elements can not be changed from person to person. The set is represented using a capital letter.

Elements of a set

The items present in a set are called elements or members of a set. Elements of a set are always enclosed in curly brackets and separated by commas. All the elements should be written in small letters.

Let us take an example:

X = {4, 8, 12, 16, 20}

X is the set and 4, 8,12,16, 20 are the elements of the set. The elements which are written in the set cannot be repeated but can be in any order.

For more detailed information, visit the Cuemath website.

Types of Set

  1. Empty Set

A set that has no element is called an empty set or void set or null set. It is denoted by { }.

Example: Set A = {}.

  1. Singleton Set

A set that has only a single element is called a singleton set.

Example: Set A = {2}.

  1. Finite set

A set that consists of a countable number of elements is called a finite set.

Example: A set of prime numbers less than 10 can be written as follows.

A = {2,3,5,7}

  1. Infinite set

Infinite set is a type of set which has an infinite number of elements.

Example: A set of all numbers which are multiples of 5 can be written as follows.

A = {5,10,15,20,25,……}

  1. Equivalent set

Two Sets are called equivalent sets if they have the same number of elements, though the elements are different.

  1. Equal set

Two sets having the same numbers of elements are called equal sets.

  1. Disjoint Sets

The two sets X and Y are said to be disjoint if there are no common elements between them.

  1. Subsets and Supersets

A set ‘X’ is said to be a subset of Y if every element of X is present in set Y, denoted as X ⊆ Y.

Example: X = {1,2,3}

Then {1,2} ⊆ X.

Y ⊇ X denotes that set Y is the superset of set A.

  1. Universal Set

A collection of all the elements of other sets is called universal sets. It is represented as ‘U.’

For Example: set X = {1,2,3}, set Y = {3,4,5,6} and Z = {5,6,7,8,9}

then, Universal set U as,

U = {1,2,3,4,5,6,7,8,9,}

Order of Sets

The order of a set means the total number of elements in a set. It denotes the size of a set. The order of sets is also called cardinality.

One important parameter to define a set is that all the elements or members of a set should be related to each other and also have a common feature.

Notation

Sets are notified by simply grouping each element separated by a comma and then putting some curly braces {} around the whole thing.

For example, {5,6,7} or {x,y,z} or {Bat, Ball, Wickets}.

The three set notations for representing sets are:

  • Statement form
  • Roster form
  • Set builder form

Statement Form

A well-defined description to show what are the elements of a set.

For example, A is the set of the first 4 even numbers.

In statement form, it can be written as {first 4 even numbers}.

Roster Form

The most common form which is used to represent sets is the roster notation in which all the elements of the sets are enclosed in curly brackets and separated by commas. For example, Set A = {3,5,7,9,11}, which is the collection of the first five odd numbers. For more detailed information about Sets visit Cuemath website.

Set Builder Form

The set builder form uses a vertical bar, with text which describes the character of the members of the set. For example, X = { a | a is an even number, a ≤ 10}. Here, the statement says all the elements of set X are even numbers and are also less than or equal to 10.

On the Cuemath website, you can get detailed knowledge of sets with examples.

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