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Set Theory is a branch of Mathematics that deals with the study of sets, their symbols, properties, and operations. Set Theory was proposed by Georg Cantor, a German mathematician.
- Set is a well-defined collection of elements in Mathematics.
- Elements are the objects or items present in a set which can be numbers, alphabets, variables, etc.
- A set is represented by a capital letter and curly brackets such as A = {...}.
- Set Theory Symbols are used in Operations on Sets such as Union and Intersection of Sets, Difference of Sets, etc.
Sets are considered to be invaluable tools for discussing some of the most complicated mathematical structures.
Read More: NCERT Solutions for Class 11 Maths Sets
Key Terms: Set Theory Symbols, Set Theory, Sets, Set Operations, Universal Set, Elements, Union of Sets, Intersection of Sets
What is Set Theory?
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Set Theory is a branch of mathematical logic that deals with Sets, which can be defined as an organized collection of objects.
- Sets are collections of elements like numbers, variables, alphabets, etc.
- The elements of a set are fixed and cannot be changed.
- Elements of a set can be any number, name, or object.
- Sets are denoted by a capital letter and the elements are added within curly brackets.
- Sets are classified into various types such as Finite and Infinite Sets, Universal Sets, Empty Sets, Singleton Sets, Equal Sets, etc.
- Set Theory is an integral component in the mathematical classification and organization of data.
Example of Sets
Given below are a few examples of Sets in Maths:
- Set of First Five Prime Numbers: Set A = {2, 3, 5, 7, 11}
- Set of Multiples of 2: Set A = {2, 4, 6, 8 ....} )
- Set of First 10 Natural Numbers: A = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
History of Set Theory
- Set Theory was formulated by a Greek mathematician named Georg Cantor between 1874 and 1897.
- He proposed it during his research about some factual problems, related to specific types of infinite sets of real numbers.
- According to the theory, a set is a collection of some definite and distinct objects of observation as a whole.
- All these objects are called members or elements of the set.
- Cantor’s theory is based on the property of the combination of real algebraic numbers.
Read More:
| Relevant Concepts | ||
|---|---|---|
| Set Formula | Equal and Equivalent Sets | Data Sets |
| Union of Sets | Venn Diagrams | Reflexive Relation |
Set Theory Symbols
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Set Theory Symbols are the various symbols used during Operations on Sets. Set Operations include the various mathematical operations that are applicable to sets such as union, intersections, difference, and the complement of a set, etc.
Assume a Universal set (U) = {1, 2, 7, 9, 13, 15, 21, 23, 28, 30}.
| Symbol | Symbol Name | Meaning | Example |
|---|---|---|---|
| { } | Set | Collection of Elements | A = {1,7,9,13,15,23}; B = {7,13,15,21} |
| A U B | Union | Elements Belonging to Either Set A or Set B | A U B = {1,7,13,15,21,23} |
| A ∩ B | Intersection | Elements Belonging to Both Set A and Set B | A ∩ B = {7,13,15} |
| A ⊆ B | Subset | It has few or all elements that are equal to the Set. | {7,15} ⊆ {7,13,15,21} |
| A ⊄ B | Not Subset | The left set is not a subset of the right one. | {1,23} ⊄ B |
| A ⊂ B | Strict Subset/Proper Subset | It has fewer elements than the Set. | {7,13,15} ⊂ {1,7,9,13,15,23} |
| A ⊃ B | Strict Superset/Proper Superset | Set A has more elements than Set B. | {1,7,9,13,15,23} ⊃ {7,13,15} |
| Ø | Empty Set | Ø = { } | C= {0} |
| A ⊇ B | Superset | Set A has more or coequal elements to Set B. | {1,7,9,13,15,23} ⊃ {7,13,15,21} |
| A ⊅ B | Not Superset | The set X is not a superset of Y. | {1,2,5} ⊅ {1,6} |
| P (C) | Power Set | All the subsets of C. | C= {4,7}, P (C) = {{}, {4}, {7}, {4,7}} |
| A = B | Equality | Both sets have the same members. | {7,13,15} = {7,13,15} |
| Ac | Complement | All objects/elements that do not belong to Set A. | U= {1,2,7,9,13,15,21,30}; Ac = {2, 21, 28, 30} |
| A \ B or A-B | Relative Complement | Elements/Objects that belong to Set A and not to B. | {1,9,23} |
| A ∆ B | Symmetric Difference | Objects that belong to A or B, however, not to their intersection. | A ∆ B = {1, 9, 21, 23} |
| a ∈ B | Element Of | Set Membership | B = {7,13,15,21},13 ∈ B |
| (a, b) | Ordered Pair | Collection of 2 Elements | (1,2) |
| x∉A | Not Element Of | There is No Set Membership. | A= {1,7,8,13,15,23}, 5 ∉ A |
| |B|, #B | Cardinality | Number of Elements of Set B | B= {7,13,15,21}, |B|=4 |
| A x B | Cartesian Product | Set of all Ordered Pairs from Set A & B | {3,5}×{7,8} = {(3,7); (3,8),(5,7),(5,8) } |
| N1 | Natural Numbers/Whole Numbers Set (Without Zero) | N1 = {1, 2, 3, 4, 5…} | 6 ∈ N1 |
| N0 | Natural Numbers/Whole Numbers Set (With Zero) | N0 = {0, 1, 2, 3, 4…} | 0 ∈ N0 |
| Z | Integer Numbers Set | Z = {….-3, -2, -1, 0, 1, 2, 3…} | -6 ∈ Z |
| Q | Rational Numbers Set | Q = {x |x = a/, a, b € Z} | 2/6 ∈ Q |
| C | Real Numbers Set | R= {x | -∞ < x <∞} | 6.343434 ∈ R |
| R | Complex Numbers Set | C= {z| z = a +bi, -∞<a<∞, -∞<b<∞} | 6+2i ∈ C |
Basic Concepts of Set Theory
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Set Theory includes various topics such as the representation of a set, types of sets, operations on sets, cardinality of a set and relations, etc. Given below are some of the basic concepts involved in set theory:
Universal Set
Universal Set contains all the elements of other sets including the own elements of the set. Universal Set is represented with the capital letter ‘U’. Sometimes, it can also be represented by the symbol ε (epsilon).
- U = {Counting Numbers}
- U = Set of Integers
Complement of Set
Complement of a Set contains all the elements of the universal set except the elements of the set under consideration. For example, if A is a set then the complements of set A will contain all the elements of the given universal set (U), which are not included in A. The complement of a set is denoted or represented as A' or Ac.
| A’= {x € U: x ≠ A} |
Set Detailed Video Explanation
Set Builder Notation
Examples of Notation of a set in Set Builder Form are as follows:
Suppose, A is a Set of Real Numbers.
| A= {x: x∈R} |
Where x belongs to all real numbers.
If A is the set of natural numbers
| A= {x: x>0} |
Set Theory
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Types of Sets
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Sets are classified into various categories depending on the elements included. The types of sets are listed below:
- Finite Set: In a finite set, the number of elements is finite or countable.
- Infinite Set: In an infinite set, the number of elements is infinite or uncountable.
- Empty Set: An empty set has no elements. It is also called Null Set.
- Unit Set: A unit set contains only one element in it. It is also called Singleton Set.
- Equal Set: If two sets have the same element, then, they are called equal sets.
- Power Set: A power set is a set of all the subsets that a set could contain.
- Universal Set: A type of set that contains all the sets under consideration is called a universal set.
- Subset: If all the elements of Set X belong to Set Y then set X is a subset of set Y.
Representation of Sets
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Sets can be represented in various manners depending on the way the elements are listed. There are three set notations or representations used for representing sets which are as follows:
- Semantic Form
- Roster Form
- Set Builder Form
Semantic Form
Semantic Form of representation of sets includes a statement to show what are the elements of a set. For instance
| ‘Set A is the list of the first five natural numbers.’ |
Roaster Form
In Roster Form, all elements or objects of a set are listed conversely separated by the commas and enclosed as curly braces {}.
Example: Let’s consider a set containing the leap years between the years 1995 and 2015. The set will be represented as:
| A= {1996, 2000, 2004, 2008, 2012} |
Set Builder Form
In Set Builder Form, it is a must that all the objects should have the same properties. There should be a certain rule or a statement that describes the common feature of all the elements of a set. It involves a vertical bar (|) or ":" in its representation, and a text describing the character of the elements of the set.
Example: Suppose Set P has all the elements that are even prime numbers. Thus,
| P = {a: a is an Even Prime Number} |
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Set Theory Solved Examples
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Here are some solved examples on Sets using Set Theory Symbols to get a better idea about the same.
Example 1: If A and B are two finite sets such that n(A)= 10, n(B) = 18 & n (A U B) = 26. Find out n(A ∩ B).
Solution: It is known that,
n (A U B) = n(A) + n(B) – n(A ∩ B)
So, n(A ∩ B)= n(A) + n(B)- n(A U B)
= 10 + 18 – 26
= 28 - 26
= 2
Thus, n(A ∩ B) is equal to 2.
Example 2: If A = {x: x is a Natural Number & Factor of 16} whereas B = {x: x is a Natural Number & It is less than 8}. Find out A U B.
Solution: According to the question, the sets will be
- A = {1,2,3,6,9,16}
- B = {1,2,3,4,5,7}
Hence, A U B = {1,2,3,4,5,6,7,9,16}
Applications of Set Theory
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Set Theory has a wide variety of applications in Mathematics, Logic, and related fields. It is used in
- Graphs
- Vector Spaces
- Ring Theory
The above-mentioned concepts can be defined as sets satisfying specific properties or axioms of sets. Set Theory is also considered to be the foundation for numerous topics such as
- Topology
- Mathematical Analysis
- Discrete Mathematics
- Abstract Algebra
Things to Remember
- Set Theory is a mathematical theory that studies Sets, their properties, operations, and representation.
- Set is defined as a well-organized collection of elements or objects in Mathematics.
- Elements of a set are the objects or items present in a set that can be any number, name, or object.
- A set is represented by a capital letter with the elements listed in a curly bracket.
- Set Theory Symbols are the symbols used during operations on sets.
- Operations on sets are union, intersection, complement, difference, and cartesian product of a set.
- Sets are expressed in three forms namely set-builder form, roster form, and statement form.
- Types of Sets in Maths are Finite Sets, Infinite Sets, Empty Sets, Singleton Sets, Universal Sets, Equal Sets, etc.
- A universal set consists of all the elements under consideration for a specific circumstance.
- Venn diagrams are used for representing the sets as well the relationship between them.
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Previous Years’ Questions
- If A and B are non-empty sets such that… (KEAM)
- Let A and B be two sets then (A∪B)′… (BITSAT - 1990)
- Two finite sets have m and n elements. The total number…
- Which of the following sets is a finite set…
- For any two sets A and B, A-(A−B) equals… (WBJEE - 2009)
- The number of proper subsets of a set having… (COMEDK UGET - 2014)
- If A and B are two sets, then A ∩ (A∪B)...
- Let S be a set containing n elements and we select two subsets… (BITSAT - 2005)
- If A and B are not disjoint sets, then n(A∪B)...
- Two sets A and B are as under: A = {(a,b)... (JEE Main - 2018)
Sample Questions
Ques. What are Venn Diagrams? (3 Marks)
Ans. Venn Diagrams refer to the pictorial representation of sets in which each set is represented as a circle.
- The elements of a set are present within the circle.
- The universal sets are represented usually by a rectangle and its subsets are represented by circles.
- Most of the relationships between sets can be represented through Venn diagrams.
Ques. What are the Set Theory Formulas? (3 Marks)
Ans. These are six formulas of the Set Theory which are as follows:
- n ( P ∪ Q ) = n(P) + n(Q) – n ( P ∩ Q)
- If P ∩ Q = ∅, then n ( P ∪ Q ) = n(P) + n(Q)
- n( P – Q) + n( P ∩ Q ) = n(P)
- n( Q – P) + n( P ∩ Q ) = n(Q)
- n( P – Q) + n ( P ∩ Q) + n( Q – P) = n ( P ∪ Q )
- n ( P ∪ Q ∪ R ) = n(P) + n(Q) + n(R) – n ( P ∩ Q) – n ( Q ∩ R) – n ( R ∩ P) + n ( P ∩ Q ∩ R)
Ques. Define Subsets. (2 Marks)
Ans. A set is called the subset of the other set when it has some or all the elements of the other set. Moreover, it should not have any elements of the supersets. A proper subset is a subset that has all the elements of a superset.
- For example, A = {4,5,6} and B = {0,1,2,3,4,5,6,7,8,9}
- A ⊆ B, as all the elements in set A are present in set B.
- B ⊇ A indicates that set B is the superset of set A.
Ques. What are the basic concepts of Set Theory? (3 Marks)
Ans. The basic concepts of the Set Theory are:
- A set is a collection of objects with actual properties.
- The objects of the sets are called the members or elements of the set.
- The uppercase letters are used to represent the sets
- Lowercase letters are used to denote the elements of the sets.
- Set theory is used to define several mathematical structures.
Ques. What are the applications of Set Theory? (3 Marks)
Ans. Set theory has numerous applications in several different fields. Given below are some major applications of set theory:
- Set theory is used in Graphs, vector spaces, ring theory, etc.
- It is the foundation of topics like topology, mathematical analysis, discrete mathematics, abstract algebra, etc.
- Set theory is used in probability.
- It is used for building more complex mathematical structures.
Ques. Define Empty Set. (2 Marks)
Ans. Empty Set is a set that doesn’t contain any element. A set of oranges in the basket of apples is an example of an empty set as there will be no oranges in the apple basket. An empty set is denoted by the symbols { } or Ø.
Ques. What is the symbol for Union of Sets? (3 Marks)
Ans. The symbol used for the Union of Sets is ‘∪’. The union of two sets can be represented as X ∪ V. It should have elements of both X and V. The total shaded area that has been covered by set X and V falls under X ∪ V.
Union of Sets is one of the fundamental set operations through which sets can be combined and are related to each other. It basically lists the elements in set A and set B or the elements in both set A and set B.
Example: {3,4} ∪ {1, 4} = {1, 3, 4}
Ques. What is Cartesian Product of Sets? (3 Marks)
Ans. If set A and set B are two given sets, then the cartesian product of A and B will be the set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. Cartesian Product of Sets A and B is denoted by A × B.
A × B = {(a, b) : a ∈ A and b ∈ B}
Assume Set A = {1,2,3} and Set B = {Bat, Ball}. Therefore,
A × B = {(1, Bat), (1, Ball), (2, Bat), (2, Ball), (3, Bat), (3, Ball)}
Ques. What is a Set? (3 Marks)
Ans. Set is defined as an organized collection of objects or elements.
- It is made up of elements which could be numbers, letters, shapes, etc.
- It is represented by a capital letter like A = {...}.
- The elements are represented using a curly bracket in a set like A = {1, 2, 3, 4, 5}.
Ques. What are Disjoint Sets? (2 Marks)
Ans. Disjoint Sets are the two sets that contain no same element.
Consider, A = {1,2,3} and B = {4,5,6}.
Here, Set A and B are disjoint sets as they have all different elements.
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