Introduction to Sets

Sets in mathematics are simply collections of distinct objects that form a group. In this step-by-step guide, you will learn more about the definition of sets.

Introduction to Sets

Step-by-step guide to sets

In mathematics, a set is a well-defined set of objects. Sets are named and displayed in capital letters. In set theory, the elements that make up a set can be anything: people, alphabets, numbers, shapes, variables, etc.

Elements of a set

The items in a set are called elements or members of a set. The elements of an array are enclosed in curly braces, separated by commas. To indicate that an element is in a set, the symbol ‘\(∈\)’ is used. If an element is not a member of a set, it is indicated by the symbol ‘\(∉\)’.

Cardinal number of a set

The cardinal number, cardinality, or order of a set represents the total number of elements in the set. Sets are defined as sets of unique elements. One of the important conditions for defining a set is that all elements of a set must be related to each other and have a common feature.

Representation of sets

Different set symbols are used to represent collections. They differ in how the elements are indexed. The three collection symbols used to represent collections are:

Semantic form:
A semantic symbol describes an expression to indicate what the elements of a set are. For example, set \(B\) is a list of five prime numbers.

Roster form:
The most common form used to represent sets is roster notation, where the elements of the sets are enclosed in curly brackets separated by commas. For example, Set \(B=\){\(2,4,6,8,10\)}, which is the collection of the first five even numbers. If there is an endless list of elements in a set, then they are defined using a series of dots at the end of the last element.

Set builder form:

A set builder symbol has a special rule or expression that specifically describes the common property of all elements of a set. The set builder form uses a vertical bar in its display with text that describes the character of the collection elements.

For example, \(A=\) { \(k | k\) is an odd number, \(k≤ 10\)}. The statement says, all the elements of set \(A\) are odd numbers that are less than or equal to \(10\). Sometimes a “\(:\)” is used in the place of the “\(|\)”.

What are the types of sets?

A set has many types, such as;

  • Empty set or null set: It has no element present in it. For example: \(A=\){} is a null set.
  • Finite set: It has a limited number of elements. For example: \(A=\){\(2,3,4,5\)}.
  • Infinite set: It has an infinite number of elements. For example: \(A=\) {\(x: x\) is the set of all whole numbers}.
  • Equal set: Two sets that have the same members. For example: \(A=\){\(3,4,5\)} and \(B=\){\(4,5,3\)}, Set \(A =\) Set \(B\).
  • Subsets: A set ‘\(A\)’ is called to be a subset of \(B\) if each element of \(A\) is also an element of \(B\). For example: \(A=\){\(3,4\)}, \(B=\){\(1,2,3,4\)}, then \(A ⊆ B\).
  • Universal set: A set that consists of all elements of other sets present in a Venn diagram. For example: \(A=\){\(1,4\)}, \(B=\){\(4,3\)}. The universal set here will be, \(U=\){\(1, 4,3\)}.

Sets formulas

Sets are used in algebra, statistics, and probability. There are some important set formulas are listed below. For both overlapping sets \(A\) and \(B\),

  • \(\color{blue}{n\left(A\:U\:B\right)=\:n\left(A\right)\:+\:n\left(B\right)-\:n\left(A\:∩\:B\right)}\)
  • \(\color{blue}{n\:\left(A\:∩\:B\right)=n\left(A\right)+n\left(B\right)-n\left(A\:U\:B\right)}\)
  • \(\color{blue}{n\left(A\right)=n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(B\right)}\)
  • \(\color{blue}{n\left(B\right)=\:n\left(A\:U\:B\right)+n\left(A\:∩\:B\right)-\:n\left(A\right)}\)
  • \(\color{blue}{n\left(A\:-\:B\right)=n\left(A\:U\:B\right)-n\left(B\right)}\)
  • \(\color{blue}{n\left(A\:-\:B\right)=\:n\left(A\right)-\:n\left(A\:∩\:B\right)}\)

For any two sets \(A\) and \(B\) that are disjoint,

  • \(\color{blue}{n\left(A\:U\:B\right)=n\left(A\right)+\:n\left(B\right)}\)
  • \(\color{blue}{A ∩ B = ∅}\)
  • \(\color{blue}{n\left(A\:-\:B\right)=n\left(A\right)}\)

Sets – Example 1:

 If set \(A=\) {\(a,b,c\)} and set \(B=\) {\(a,b,c,l,z,r\)}, find \(A ∩ B\).

Solution:

\(A∩B=\) {\(a,b,c\)}

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