Origin of Set Theory
Georg Cantor (1845-1918), a German mathematician, was the first to propose the concept of ‘Set Theory.’ While researching “Problems on Trigonometric Series,” he came across sets, which have since become one of mathematics’ most fundamental ideas. It will be impossible to explain other concepts such as relations, functions, sequences, probability, geometry, and so on without first comprehending sets.
Concept of Set Theory in Mathematics
The concept of sets in mathematics deals with the properties and operations on collections of objects. This is particularly important for classification, organization, and is the base for many forms of data analysis.
In mathematics, sets are essentially a collection of different items that form a group. A set can contain any number of elements, such as numbers, days of the week, car types, and so on. Each object in the set is referred to as an element of the set. When writing a set, curly brackets are used. This is an example of a set in its most basic form Set A = {1,2,3,4,5}.
Important Sets used in Mathematics
N: It consists of Set of all natural numbers = {1, 2, 3, 4, …..}
Q: Set of all rational numbers
R: Set of all real numbers
W: Set of all whole numbers
Z: It contains Set of all integers = {….., -3, -2, -1, 0, 1, 2, 3, …..}
Z+: Set of all positive integers
What are the Various Types of Sets?
For Example: A set of natural numbers up to 10. A = {1,2,3,4,5,6,7,8,9,10}
For Example: A set of all natural numbers. A = {1,2,3,4,5,6,7,8,9……}
A set of apples in a basket of grapes is an example of an empty set because there are no apples in a grape basket.
Example: In a basket of grapes, there is only one apple.
For instance : A = {1,2,3,4} ; B = {4,3,2,1}.
A = B
(A) n(A) n(A) n(A) n(A) (B)
where A and B are two distinct sets of the same number of elements
Assume A = 1,2,3,4 and B = Red, Blue, Green, Black.
Set A has four elements and set B has four elements as well. As a result, sets A and B are equivalent.
Example: If A = {1,2,3} and B {2,3,4,5}, universal set is, U = {1,2,3,4,5}
For instance: A = {1,2,3}
So, {1,2} ⊆ A.
Likewise, other subsets of set A are: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.
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Three Methods of Describing Sets
There are mainly three methods of representing the elements within a set. They are enlisted below:
1. Statement Form: In the statement form, the accurate description and properties of a member of a set are written and enclosed within curly brackets.
For instance, the set of even numbers is less than 15.
In the statement form method, it is represented as {even numbers less than 15}.
2. Roster Form: In the roster form of describing a set, each element of the set is listed within the curly brackets, with a comma for the separation of elements.
For instance, the set of natural numbers is less than 5.
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,……….
Natural Number less than 5 is equal to 1, 2, 3, 4
As a result, the set is N = { 1, 2, 3, 4 }
3. Set Builder Form: The set builder notation begins with an alphabet, say x, as a variable, followed by a colon. Then all the properties that an element x must satisfy to be considered a member of the set are then written. This notation is perfect to state all the properties of the elements of a particular set.
For Example: Write the following sets in set builder form: A={2, 4, 6, 8}
Solution:
2 = 2 x 1
4 = 2 x 2
6 = 2 x 3
8 = 2 x 4
So, the set builder form is A = {x: x=2n, n ∈ N and 1 ≤ n ≤ 4}
Set Theory
The essential features of the set theory include:
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Making of a set involves the grouping of objects of any kind into a single entity
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The specific relationship that may or may not exist between an object and a set is called a membership relationship. An object is a member of a set or it is not; there is no in-between.
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The Principle of Extension states that the set is defined by its elements instead of a single means of defining the group. Thus, sets P and Q are equal only if all the elements they contain intersect, and none of them have any unique elements which are not present in the other set.
Sets and Relations Formulae
The set theory formulas are listed below. For any three sets P, Q, and R:
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n ( P ∪ Q ) = n(P) + n(Q) – n ( P ∩ Q)
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If P ∩ Q = ∅, then n ( P ∪ Q ) = n(P) + n(Q)
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n( P – Q) + n( P ∩ Q ) = n(P)
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n( Q – P) + n( P ∩ Q ) = n(Q)
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n( P – Q) + n ( P ∩ Q) + n( Q – P) = n ( P ∪ Q )
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n ( P ∪ Q ∪ R ) = n(P) + n(Q) + n(R) – n ( P ∩ Q) – n ( Q ∩ R) – n ( R ∩ P) + n ( P ∩ Q ∩ R)
Properties of Sets
Commutative Property: When calculating the union or intersection of a set, changing the order of sets does not change the answer. Just like 2+8=8+2=10, even the following statements are true.
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P∪Q = Q∪P
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P∩Q = Q∩P
Associative Property: In an expression having two or more numbers or variables in a row of the same relational operators, the sequence in which the operations are performed does not make any difference in the result as long as the sequence of the operands is not changed. Just like (3+6)+4=(6+4)+3=13, the following statements also hold true.
Distributive Property: Distributive law is when the operation is rearranged logically to arrive at the same results. For example, in arithmetic, 3x(2+8)=(3×2)+(3×8)=30, the same property is seen in set theory.
De Morgan’s Law: De Morgan’s Law holds that the complement of the intersection of the two sets is the union of their complements and the complement of the union of the two sets is the intersection of their complements.
Complement Law: In set theory, the complement of set P refers to every element that is not present in set P. When all of the sets existing in the world are assumed to be subsets of a given set
R, the absolute complement of A is the group of elements that is present in R but absent in A.
Idempotent Law and Law of Null and Universal Set: A idempotent element is an element, which when multiplied by itself, gives itself as the result. For example, 1 is idempotent multiplication.
For any finite set P
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P ∪ P = P
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P ∩ P = P
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∅’ = U
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∅ = U’
This is all about sets, the concept behind different kinds of sets, and their properties. Focus on the new concepts and learn how to define different kinds of sets with examples to understand clearly.