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Description

Set Theory is a branch of mathematics that teaches us about sets and their properties. A collection of objects or groups of objects is referred to as a set. These objects are frequently referred to as elements or members of sets. A set is, for example, a group of cricket players. Because a cricket team can only have 11 players at a time, we can say that this set is finite. 

A set of English vowels is another example of a finite set. There are, however, many sets with infinite members, such as a set of natural numbers, a set of whole numbers, a set of real numbers, a set of imaginary numbers, and so on.

Set Theory Origin

Georg Cantor (1845-1918), a German mathematician, pioneered the concept of “theory of sets” or “Set Theory.” While working on “Problems on Trigonometric Series,” he came across sets, which have since become one of mathematics’ most fundamental concepts. It will be difficult to explain other concepts such as relations, functions, sequences, probability, geometry, and so on if you do not understand sets. At Cuemath, we have a fantastic team of educators and teachers who are committed to helping students understand the true value of math. 

What Are Sets?

A set is defined in mathematics as a collection of well-defined objects. For example, the set of natural numbers ranging from 1 to 10, or the set of even numbers ranging from 1 to 20. If we change the order in which we write the elements in a set, nothing changes in the set. When one or more elements of a set are repeated, the set remains unchanged. In this article, we will look at some of the most important properties of set operations.

If A and B are two sets, we can define a set A intersection B, denoted by AB, whose elements are all the common elements of A and B. Another set A union B, denoted by AB, is the set that contains all of A and B’s elements.

Representation of Sets

We can represent sets in two ways:

  1. Roster Form or Tabular form
  2. Set Builder Form

What is a Roster Form?

All of the set’s elements are listed in roster form, separated by commas and enclosed by curly braces.

For example, if set represents all of the leap years between 1995 and 2015, it would be written in Roster form as: 

A ={1996,2000,2004,2008,2012}

The elements between the braces are now written in ascending order. This could be in any order, descending or random. As previously stated, the order of a set represented in the Roster Form is irrelevant.

Furthermore, multiplicity is ignored when representing the sets. For example, if L represents a set containing all of the letters in the word ADDRESS, the correct Roster form representation would be 

L ={A,D,R,E,S }= {S,E,D,A,R}  

L≠ {A,D,D,R,E,S,S}

What is Set Builder Form?

Set-builder notation is a mathematical notation used in set theory and its applications to logic, mathematics, and computer science to describe a set by enumerating its elements or stating the properties that its members must satisfy.

Sets of Various Kinds

The sets are further classified into various types based on the elements or types of elements. In basic set theory, the various types of sets are as follows:

  1. The number of elements in a finite set is finite; it is known as a finite set.
  2. The number of elements in an infinite set is infinite, it is known as an infinite set.
  3. It is an empty set because it contains no elements.
  4. It is a singleton set because it only has one element.
  5. Set of equals: If two sets have the same elements, they are equal.
  6. Sets that are equivalent: If two sets have the same number of elements, they are equivalent.

A power set is a collection of every possible subset.

  1. Any set that contains all of the sets under consideration is referred to as a universal set.
  2. When all of the elements of set A belong to set B, then set A is a subset of B.
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