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

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, Black. Both 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.

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,

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 

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 A^C is 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 A^C

Ans: A^C = U - A = {1, 3, 5, 7, 9, 10}

(V) Cartesian Product of Sets

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.