Sets

Sets Theory

Q1 Define Sets

Collection of things are  that are related and well defined are called sets. For example
list of fruits, vegetables, sports equipment, natural numbers etc

Q2 What are the different types of sets?

a) Universal Sets
b) Equal Sets
c) Null Sets
d) Subsets
e) Equivalent Sets
f) Empty Sets
g) Singleton Sets
h) Infinite Sets
i) Finite Sets

Q 3 How do we name a set?
Ans Sets are denoted by capital letter
A={1,2,3}={3,2,1}
Order of elements is not important here.

Q4 Is repetition of elements or member in the set allowed?
Ans No, repeat elements are not allowed in set. The single of each repeated element will be considered.
To write set for word school
B={s,c,h,o,l} is correct
B={s,c,h,o,o,l} is incorrect

Q5 Write the different forms of Sets.
Ans a) List form and Set builder form

Q6 Write a set of first 5 natural numbers.
Ans {1,2,3,4,5}. Here number inside the curly braces are known as elements or
members. It is one of the most common method of listing elements.

Q7 Write a set of first 5 natural numbers in another way?
Ans {x:x < 6, x  N}

Q8 What are finite and infinite sets?

Ans Let us take an example to understand
{...,-3,-2,-1,0,1,2,3...} is a infinite set
List of subjects in class XII
{maths, english, physics, chemistry, biology} is a finite set.
set of week days is a finite set
set of real numbers.

Q9 What are Singleton and Empty sets?

Ans Set with one element is called Singleton.
 Set with no element is called empty set.
List of prime numbers which is even
{2} only one element hence it is called singleton set.
List the colors that comes after 7 colors in rainbow
{void}

Q10 What is  a cardinal number?

Ans The number of elements or members is called cardinal number.
A={1,2,3,4,5}
number of elements is 5
n(A)=3
Cardinality is also know as the size of the set. 

Q11 What are equivalent sets?

Ans Any two sets are equivalent  if they have the same cardinal number
A={1,2,3} B={3,5,6}
n(A)=n(B)=3
Here sets A and B are equivalent

Q12 What are equal sets?

Ans Two sets are equal if they have precisely the same elements or members.
order does not matter.
A={5,6,7} B={5,6,7}
C={6,7,5}
Here A=B
B=C and C=A
Let us take another case
D={1,2,3} E={1,2,3,4}
As n(D) is not equal to n(E)
Here D is not equal to E

Q 13 Are all equal sets are equivalent?
Yes

Q 14 Are all equivalent sets are equal?
Ans Not always equal.

Q 15  Write an example of list form.
Ans  List first five natural numbers
{1,2,3,4,5}

Q 16 Write an example of set builder method.
{x:x < 6,xN} it is read as " All x such that x is less than 6, x belongs to natural numbers." 

Q 17 What are subsets?

Ans Let understand subsets by an example

A={1,2} B={1,2,3}
Here A is a part of B hence it is a subset of B. 
By definition every element in A is also present in B.
Every set is a subset of itself
A   A
B   B
It can also be written as A  B

Q 18 What are proper subsets or strict subsets?

Ans A={1,4,5} B={1,4,5.6,8}
Here all the elements of A are present in B but there are two elements of a B that are not present in A. Hence  A  B
C={1,2,3} D={1,2,3}
Here all elements of C are present in D but there is not even one element of D which is present in D but not in C. Hence by definition  C is not a proper subset of D
C D

Q 19 What are supersets?

Let us take an example
A={1,4,5} B={1,4,5.6,8}
Here B is a superset of A as it contains all the elements of A
BA
Q 20 What is a null set?
Ans A set with no elements  denoted by 
A={1,2,3}


Q 21 What is a Universal Set?

Let us take an example
A={1,2,3} B={4,5,6} C={7,8,9}
U={1,2,3,4,5,6,7,8,9}

Here U is a universal sets as all other sets are subsets of this universal set.

Q 22 What are Venn diagram?
Ans
Set of students like maths A={John, Sita, Pranav}
Set of students like english B={Mary, Babita, Charan}

Q23 How do you represent Union, intersection of sets through venn diagram?

Union = 


Intersection=

Q24 What do you mean by well-defined sets?

Well-defined means, it must be absolutely clear that which object belongs to the set and which does not.

Some common examples of well defined sets:

• The collection of vowels in English alphabets. This set contains five elements, namely, a, e, i, o, u
• N = {1,2,3,…} is the set of counting numbers, or naturals.
• N = {1,2,3,…} is the set of counting numbers, or naturals.
• Z = {…,−3,−2,−1,0,1,2,3,…} is the set of integers.

Q25 Define Intersection of Sets.

The intersection of sets A and B, denoted as A ∩ B, is the set of elements common to both A AND B.

For example:

A = {1,2,3,4,5}

B = {2,4,6,8,10}

The intersection of A and B (i.e. A∩B) is simply {2, 4}

Q 26 Define Union of Sets

The union of sets A and B, written as A∪B, is the set of elements that appear in either A OR B.

For example:

A = {1,2,3,4,5}

B = {2,4,6,8,10}

The union of A and B (i.e. A∪B) is {1, 2, 3, 4, 5, 6, 8, 10}

Q27 What do you understand by difference of Sets?

The difference of sets A and B, written as A-B, is the set of elements belonging to set A and NOT to set B.

For example:

A = {1,2,3,4,5}

B = {2,3,5}

The difference of A and B (i.e. A-B) is {1,4}

NOTE: A-B ≠ B-A

Q 28 What do you mean by Cartesian Product of Sets? How it is performed?

The Cartesian product of sets A and B, written A x B, is expressed as:

A x B = {(a,b)│a is every element in A, b is every element in B}

For example:

A = {1,2}

B = {4,5,6}

The Cartesian product of A and B (i.e. A x B) is {(1,4), (1,5), (1,6), (2,4), (2,5), (2,6)}

Now, lets us try doing some questions based on Set Theory.

 

Q 29: If ∪ = {1, 3, 5, 7, 9, 11, 13}, then which of the following are subsets of U.

B = {2, 4}

A = {0}

C = {1, 9, 5, 13}

D = {5, 11, 1}

E = {13, 7, 9, 11, 5, 3, 1}

F = {2, 3, 4, 5}

Answer: Here, we can see that C, D and E have the terms which are there in ∪. Therefore, C, D and E are the subsets of ∪.

Q 30: Let A and B be two finite sets such that n(A) = 20, n(B) = 28 and n(A ∪ B) = 36, find n(A ∩ B).

Solution: Using the formula n(A ∪ B) = n(A) + n(B) - n(A ∩ B).

then n(A ∩B) = n(A) + n(B) - n(A ∪B)

= 20 + 28 - 36

= 48 - 36

= 12

 Q31 What do you mean by complement of set, explain with the help of example.

Ans  When all sets under consideration are considered to be subsets of a given set U, the absolute complement of A is the set of elements in U but not in A.

 Let U be a universal set such 
 U={1,2,3,4,5,6,7,8}
P={1,2,3}
P'={4,5,6,7,8}