Let’s think of a bag of marbles as an example. Each marble is different, and this bag contains these marbles and is in fact a set of marbles. In mathematics, this bag is said to be an ‘Set’ and the marbles that are in it are called ‘elements’ or ‘members.
There is one catch however, and that is that a set is a collection of distinct objects.” Well-defined” means that it is easy to know whether a given object is a member of the set or not.
Representing Sets
There are a couple of common ways to describe a set:
The Roster Notation:
The members of a set are written out by enumerating the elements and encasing them in curly braces {}, and the members are separated by commas. For instance, the primary colors can be described as the set of colors which include {red, yellow, blue}.
Set-Builder Notation:
Write down the set if you specify the conditions that the components must meet. For instance, the set of even numbers can be described by the following set builder notation: {x | x is even}.
Some Important Concepts Related to Set Theory
Elements:
The objects within a set. We use the symbol '∈' to denote membership. For example, 2 ∈ {2, 4, 6} reads as "2 is an element of the set {2, 4, 6}."
Cardinality:
It refers to the actuality of the members of a set. Therefore, the cardinality of the set = {2, 4, 6} is 3.
Empty Set:
A set which has no elements in it, that is, {} or Ø.
Universal Set:
The set which includes all the elements which are suitable for consideration in a specific situation.
Subsets:
A set A said to be subset of a set B (A ⊆ B) if each element of A is also an element of B. For instance, the set {1, 2} is a subset of the set {1, 2, 3}, {1, 2} ⊆ {1, 2, 3}
Proper Subsets:
If a set A is a subset of a set B and does not equal B (A ⊂ B), then A is said to be proper subset of B. This means B must contain at least an element that is not in A.
Equal Sets:
When two sets are said to be equal it means that both the sets contain the same elements.
Power Set:
The set of all the subsets of a given set. For instance, the power set of {1, 2} will be {{}, {1}, {2}, {1, 2}}.
Types of Sets
Finite Set:
A mathematical set which contains a definite and limited number of elements. The set {1, 2, 3, 4} is a finite set.
Infinite Set:
A set which contains an unlimited number of members. The set of all natural numbers is an example of an infinite set.
Operations on Sets
Union (∪):
Merges all the elements of two given sets. For instance, (1, 2) ∪ (3, 4) = {1, 2, 3, 4}.
Intersection (∩):
Produces the elements which are present in both sets. ∩ {1, 2, 3} ∩ {3, 4, 5} = {3}.
Difference (-):
Subtracts the elements of one set from the other. The result of {1, 2, 3} – {2, 3} = {1}.
Complement ('):
Gives the elements in the universal set that are not in the given set.
Why are Sets Important?
Sets are the most basic and the most important are concepts used in mathematics develop which language to count and name the collections and the objects in the collections. They are used in various fields; They are used in various fields, including:
Computer Science: Business intelligence, data modeling and organization, and programming.
Probability and Statistics: Defining the sample spaces and events.
Logic: The ability to state logical propositions as well as the relationships between them.
This introduction gives the reader a taste of what is to come in the world of sets. In due course, you will come across more advanced and fascinating ideas and usages that highlight the relevance of set theory.
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