Important Formulae: Set Theory

# Important Formulae: Set Theory | CSAT Preparation - UPSC PDF Download

 Table of contents What is a Set? Types of Sets Representation of Sets Operations on Sets Laws of Sets

## What is a Set?

A set is a collection of well-defined individual objects that form a group. A set can contain any group of items, such as a set of numbers, a day of the week, or a vehicle. Each element of the set is called an element of the set. Curly braces are used to create sets.

A very simple example of a set is: Set A = {1,2,3,4,5}. There are various notations for representing the elements of a set.

### Types of Sets

1. Finite set: This type encompasses a limited number of elements.
For instance: A set comprising natural numbers up to 10: A = {1,2,3,4,5,6,7,8,9,10}.
2. Infinite set: Within this category, the element count is limitless.
For instance: A set encompassing all natural numbers: A = {1,2,3,4,5,6,7,8,9……}.
3. Empty set: This set contains no elements.
For instance:A set denoting apples within a basket of grapes is an empty set due to the absence of apples in a grape basket.
4. Singleton set: It consists of only one element.
For instance: Set of an even prime number i.e.. {2}.
5. Equal set: When two sets share identical elements, they are equal.
For instance: A = {1,2,3,4} and B = {4,3,2,1}A equals B.
6. Equivalent set: Sets are deemed equivalent when they contain the same count of elements. This is symbolically expressed as:
(A) n(A) n(A) n(A) n(A) (B), where A and B are two distinct sets possessing an equal number of elements.
Suppose A = {1,2,3,4} and B = Red, Blue, Green, BlackBoth set A and set B possess four elements, making them equivalent.
7. Power set: This is a compilation of all possible subsets.
8. Universal set: This encompasses all sets being considered.
For instance: If A = {1,2,3} and B = {2,3,4,5}, the universal set is U = {1,2,3,4,5}.
9. Subset: Set A is said to be a subset of Set B if each and every element of Set A is also contained in Set B. Set A is said to be a proper subset of Set B if Set B has at least one element that is not contained in Set A.
For example: A = {1,2,3}.
Thus, {1,2} is a subset of A.
Additional subsets of set A include: {1},{2},{3},{1,2},{2,3},{1,3},{1,2,3},{}.
A set with ‘n’ elements will have 2n subsets (2n – 1 proper subsets)

Do you Know?

The number of elements in a set is called its cardinal number and is written as n(A). A set with cardinal number 0 is called a null set while that with cardinal number ∞ is called an infinite set.

## Representation of Sets

1. ### Statement Form

In statement form, the well-defined descriptions of a member of a set are written and enclosed in the curly brackets.
Example: Write the set of even numbers less than 15.
Solution: In statement form, it can be written as {even numbers less than 15}.
2. ### Roster or Tabular Form

In Roster form, all the elements of a set are listed.

Example: Write the set of natural numbers less than 5.
Solution:
Natural Number = 1, 2, 3, 4, 5, 6, 7, 8,………. ,
Natural Number less than 5 = 1, 2, 3, 4
Therefore, the set is N = { 1, 2, 3, 4 }.
3. ### Set Builder Form

The general form is, A = { x : property }
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}

## Operations on Sets

### (I) Union

Union of the sets A and B, denoted by A ∪ B, is the set of distinct elements that belong to set A or set B, or both.

 n(A U B) = n(A) + n(B) – n(A ∩ B)

Here,

n(A U B) = Total number of elements in A U B; is called the cardinality of a set A U B

n(A) = Number of elements in A; is called the cardinality of set A

n(B) = Number of elements in B; is called the cardinality of set B.

n(C) = Number of elements in C; is called the cardinality of set C.

If A, B and C are 3 finite sets in U then,

 n(A∪B∪C)= n(A) +n(B) + n(C) - n(B⋂C) - n (A⋂ B)- n (A⋂C) + n(A⋂B⋂C)

Example: Let U be a universal set consisting of all the natural numbers until 20 and set A and B be a subset of U defined as A = {2, 5, 9, 15, 19} and B = {8, 9, 10, 13, 15, 17}. Find A ∪ B.

Solution : We know,  n(A U B) = n(A) + n(B) – n(A ∩ B)

Given, U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20}

A = {2, 5, 9, 15, 19}

B = {8, 9, 10, 13, 15, 17}

A ∪ B = {2, 5, 8, 9, 10, 13, 15, 17, 19}

### (II) Intersection

The intersection of the sets A and B, denoted by A ∩ B, is the set of elements that belong to both A and B i.e. set of the common elements in A and B.

Example: If A = { 3, 5, 7, 9, 11}, B = {7, 9, 11, 13}, and C = {11, 13, 15}, then find B ∩ C and A ∩ B ∩ C.

Ans: Given,

A = { 3, 5, 7, 9, 11} , B = {7, 9, 11, 13} , C = {11, 13, 15}

B ∩ C = {11, 13}

A ∩ B ∩ C = {11}

### (III) Set Difference

The difference between sets is denoted by ‘A – B’, which is the set containing elements that are in A but not in B. i.e., all elements of A except the element of B.

Some of the properties related to difference of sets are listed below:

1. Suppose two sets A and B are equal then,
A – B = A – A = Ø (empty set) and B – A = B – B = Ø.
2. The difference between a set and an empty set is the set itself, i.e, A – Ø = A.
3. The difference of a set from an empty set is an empty set, i.e, Ø – A = Ø.
4. The difference of a set, say A from universal set U is equal to empty set,
i.e. A – U = Ø.
5. When a superset is subtracted from a subset, then result is an empty set,
i.e, A – B = Ø if A ⊂ B
6. If A and B are disjoint sets (no common elements for A and B),
then A – B = A and B – A = B.

Example : If X = {11, 12, 13, 14, 15}, Y = {10, 12, 14, 16, 18} and Z = {7, 9, 11, 14, 18, 20}, then find the following:
(i) X – Y – Z
(ii) Y – X – Z
(iii) Z – X – Y

Ans: Given,

X = {11, 12, 13, 14, 15}

Y = {10, 12, 14, 16, 18}

Z = {7, 9, 11, 14, 18, 20}

(i) X – Y – Z = {11, 12, 13, 14, 15} – {10, 12, 14, 16, 18} – {7, 9, 11, 14, 18, 20}

= {13, 15}

(ii) Y – X – Z = {10, 12, 14, 16, 18} – {11, 12, 13, 14, 15} – {7, 9, 11, 14, 18, 20}

= {10, 16}

(iii) Z – X – Y = {7, 9, 11, 14, 18, 20} – {11, 12, 13, 14, 15} – {10, 12, 14, 16, 18}

= {7, 9, 20}

### (IV) Complement

The complement of a set A, denoted by Ais the set of all the elements except the elements in A. Complement of the set A is U – A.

Example: Let U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {2, 4, 6, 8}. Find AC

Ans: AC = U - A = {1, 3, 5, 7, 9, 10}

### (V) Cartesian Product of Sets

If set A and set B are two sets then the cartesian product of set A and set B is a set of all ordered pairs (a,b), such that a is an element of A and b is an element of B. It is denoted by A × B.

We can represent it in set-builder form, such as:

A × B = {(a, b) : a ∈ A and b ∈ B}

Example: set A = {1,2,3} and set B = {Bat, Ball}, then;

A × B = {(1,Bat),(1,Ball),(2,Bat),(2,Ball),(3,Bat),(3,Ball)}

## Laws of Sets

(1) De Morgan’s Laws: For any two finite sets A and B;

(i) A – (B U C) = (A – B) ∩ (A – C)

(ii) A - (B ∩ C) = (A – B) U (A – C)

De Morgan’s Laws can also we written as:

(i) (A U B)’ = A' ∩ B'

(ii) (A ∩ B)’ = A' U B'

(2) Complement Laws: The union of a set A and its complement A’ gives the universal set U of which, A and A’ are a subset.

A ∪ A’ = U

Also, the intersection of a set A and its complement A’ gives the empty set ∅.

A ∩ A’ = ∅

For Example: If U = {1 , 2 , 3 , 4 , 5 } and A = {1 , 2 , 3 } then A’ = {4 , 5}. From this it can be seen that

A ∪ A’ = U = { 1 , 2 , 3 , 4 , 5}

Also

A ∩ A’ = ∅

(3) Law of Double Complementation: According to this law if we take the complement of the complemented set A’ then, we get the set A itself.

(A’)’ = A

In the previous example we can see that, if U = {1 , 2 , 3 , 4 , 5} and A = {1 , 2 ,3} then A’ ={4 , 5}. Now if we take the complement of set ‘A’ we get,

(A’)’ = {1 , 2 , 3} = A

This gives back the set A itself.

(4) Law of empty set and universal set: According to this law the complement of the universal set gives us the empty set and vice-versa i.e.,

∅’ = U And U’ = ∅. This law is self-explanatory.

Example: If U = {a, b, c, d, e, f}, A = {a, b, c}, B = {c, d, e, f}, C = {c, d, e}, find (A ∩ B) ∪ (A ∩ C).

Ans: A ∩ B = {a, b, c} ∩ {c, d, e, f}

A ∩ B = { c }

=A ∩ C = { a, b, c } ∩ { c, d, e }

A ∩ C = { c }

Then (A ∩ B) ∪ (A ∩ C) = { c }

Example: Give examples of finite sets.

Ans: The examples of finite sets are: Set of months in a year

1. Set of days in a week
2. The Set of natural numbers less than 20
3. Set of integers greater than -2 and less than 3

Example: If U = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11}, A = {3, 5, 7, 9, 11} and B = {7, 8, 9, 10, 11}, Then find (A – B)′.

Ans: A – B is a set of member which belong to A but do not belong to B

∴ A – B = {3, 5, 7, 9, 11} – {7, 8, 9, 10, 11}

A – B = {3, 5}

According to formula, (A − B)′ = U – (A – B)

∴ (A − B)′ = {2, 3, 4, 5, 6, 7, 8, 9, 10, 11} – {3, 5}

So, (A − B)′ = {2, 4, 6, 7, 8, 9, 10, 11}.

EduRev Tip: Any set is a subset of itself, but not a proper subset. The empty set, denoted by ∅ , is also a subset of any given set X. The empty set is always a proper subset, except of itself. Every other set is then a subset of the universal set.

The document Important Formulae: Set Theory | CSAT Preparation - UPSC is a part of the UPSC Course CSAT Preparation.
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## FAQs on Important Formulae: Set Theory - CSAT Preparation - UPSC

 1. What are the different types of sets in set theory?
Ans. In set theory, there are various types of sets such as finite sets, infinite sets, empty sets, singleton sets, equal sets, and equivalent sets.
 2. How are sets represented in set theory?
Ans. Sets in set theory are typically represented using curly braces { } to enclose the elements of the set, separated by commas. For example, a set containing the elements 1, 2, and 3 would be written as {1, 2, 3}.
 3. What are some common operations performed on sets in set theory?
Ans. Common operations on sets include union, intersection, difference, and complement. These operations help to combine, compare, or manipulate sets in various ways.
 4. What are the laws of sets in set theory?
Ans. The laws of sets in set theory include the commutative law, associative law, distributive law, identity law, and complement law. These laws govern how sets interact with each other when undergoing set operations.
 5. What are some important formulae in set theory for CAT exam preparation?
Ans. Some important formulae in set theory for CAT exam preparation include the formula for the number of subsets of a set (2^n), the formula for the number of elements in the union of two sets (|A ∪ B| = |A| + |B| - |A ∩ B|), and the formula for the number of elements in the intersection of two sets (|A ∩ B| = |A| + |B| - |A ∪ B|).

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