Have you ever tried to sort your wardrobe by colours or organise your playlist by mood? This is what the concept of sets teaches us in mathematics! It provides a systematic way to group and classify objects based on specific properties. From data science to probability, sets are used in every aspect of modern mathematics. The concepts like set notation, types of sets, Venn diagrams, union, intersection, complements, etc., are all going to be applied in this exercise.
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The NCERT Solutions for Chapter 1 Miscellaneous Exercise will help you break down each question in a simple and logical way. These NCERT solutions will improve your problem-solving speed and prepare you well for upcoming tests. Students can also access the NCERT notes for this chapter for feasible learning.
Question 1: Decide, among the following sets, which sets are subsets of one another:
A = { x : x $\in$ R and x satisfy $x^{2}$ – 8x + 12 = 0 }, B = { 2, 4, 6 },
C = { 2, 4, 6, 8, . . . }, D = { 6 }.
Answer:
Solution of this equation are $x^{2}$ – 8x + 12 = 0
( x - 2 ) ( x - 6 ) = 0
X = 2,6
$\therefore$ A = { 2,6 }
B = { 2, 4, 6 }
C = { 2, 4, 6, 8, . . . }
D = { 6 }
From the sets given above, we can conclude that A $\subset$ B, A $\subset$ C, D $\subset$ A, D $\subset$ B, D $\subset$ C, B $\subset$ C
Hence, we can say that D $\subset$ A $\subset$ B $\subset$ C
Question 2: (i) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If x $\in$ A and A $\in$ B , then x $\in$ B
Answer:
The given statement is false,
example: Let A = { 2,4 }
B = { 1,{2,4},5}
x be 2.
then, 2 $\in$ { 2,4 } = x $\in$ A and { 2,4 } $\in$ { 1,{2,4},5} = A $\in$ B
But 2 $\notin$ { 1,{2,4},5} i.e. x $\notin$ B
Question 2: (ii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If A $\subset$ B and B $\in$ C , then A $\in$ C
Answer:
The given statement is false,
Let , A = {1}
B = { 1,2,3}
C = {0,{1,2,3},4}
Here, {1} $\subset$ { 1,2,3} = A $\subset$ B and { 1,2,3} $\in$ {0,{1,2,3},4} = B $\in$ C
But, {1} $\notin$ {0,{1,2,3},4} = A $\notin$ C
Question 2: (iii) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If A $\subset$ B and B $\subset$ C , then A $\subset$C
Answer:
Let A ⊂ B and B ⊂ C
There is an element x such that
Let, x $\in$ A
$\Rightarrow$ x $\in$ B ( Because A $\subset$ B )
$\Rightarrow$ x $\in$ C ( Because B $\subset$ C )
Hence, the statement is true that A $\subset$ C
Question 2: (iv) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If A $\not\subset$ B and B $\not\subset$ C , then A $\not\subset$ C
Answer:
The given statement is false
Let , A = {1,2}
B = {3,4,5 }
C = { 1,2,6,7,8}
Here, {1,2} $\not\subset$ {3,4,5 } = A $\not\subset$ B and {3,4,5 } $\not\subset$ { 1,2,6,7,8} = B $\not\subset$ C
But , {1,2} $\subset$ { 1,2,6,7,8} = A $\subset$ C
Question 2: (v) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If x $\in$ A and A $\not\subset$ B , then x $\in$ B
Answer:
The given statement is false,
Let x be 2
A = { 1,2,3}
B = { 4,5,6,7}
Here, 2 $\in$ { 1,2,3} = x $\in$ A and { 1,2,3} $\not \subset$ { 4,5,6,7} = A $\not \subset$B
But, 2$\notin$ { 4,5,6,7} implies x $\notin$ B
Question 2: (vi) In each of the following, determine whether the statement is true or false. If it is true, prove it. If it is false, give an example.
If A $\subset$ B and x $\notin$ B , then x $\notin$ A
Answer:
The given statement is true,
Let, A $\subset$ B and x $\notin$ B
Suppose, x $\in$ A
Then, x $\in$ B , which is contradiction to x $\notin$ B
Hence, x $\notin$ A.
Question 3: Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C. Show that B = C.
Answer:
Let A, B, and C be the sets such that A $\cup$ B = A $\cup$ C and A $\cap$ B = A $\cap$ C
To prove: B = C.
A $\cup$ B = A + B - A $\cap$ B = A $\cup$ C = A + C - A $\cap$ C
A + B - A $\cap$ B = A + C - A $\cap$ C
B - A $\cap$ B = C - A $\cap$ C ( since A $\cap$ B = A $\cap$ C )
B = C
Hence proved that B = C.
Question 4: Show that the following four conditions are equivalent :
(i) A $\subset$ B(ii) A – B = $\phi$ (iii) A $\cup$ B = B (iv) A $\cap$ B = A
Answer:
First, we need to show A$\subset$ B $\Leftrightarrow$ A – B = $\phi$
Let A $\subset$ B
To prove: A – B = $\phi$
Suppose A – B $\neq$ $\phi$
this means, x $\in$ A and x $\not =$ B , which is not possible as A $\subset$ B .
SO, A – B = $\phi$.
Hence, A $\subset$ B $\implies$ A – B = $\phi$.
Now, let A – B = $\phi$
To prove: A $\subset$ B
Suppose, x $\in$ A
A – B = $\phi$ so x $\in$ B
Since, x $\in$ A and x $\in$ B and A – B = $\phi$ so A $\subset$ B
Hence, A$\subset$ B $\Leftrightarrow$ A – B = $\phi$.
Let A$\subset$ B
To prove : A $\cup$ B = B
We can say B $\subset$ A $\cup$ B
Suppose, x $\in$ A $\cup$ B
means x $\in$ A or x $\in$ B
If x $\in$ A
since A$\subset$ B so x $\in$ B
Hence, A $\cup$ B = B
and If x $\in$ B then also A $\cup$ B = B.
Now, let A $\cup$ B = B
To prove : A$\subset$ B
Suppose : x $\in$A
A $\subset$ A $\cup$ B so x $\in$ A $\cup$ B
A $\cup$ B = B so x $\in$ B
Hence,A$\subset$ B
ALSO, A$\subset$ B $\Leftrightarrow$ A $\cup$ B = B
NOW, we need to show A $\subset$ B $\Leftrightarrow$ A $\cap$ B = A
Let A $\subset$ B
To prove : A $\cap$ B = A
Suppose : x $\in$ A
We know A $\cap$ B $\subset$ A
x $\in$ A $\cap$ B Also ,A $\subset$ A $\cap$ B
Hence, A $\cap$ B = A
Let A $\cap$ B = A
To prove: A $\subset$ B
Suppose : x $\in$ A
x $\in$ A $\cap$ B ( replacing A by A $\cap$ B )
x $\in$ A and x $\in$ B
$\therefore$ A $\subset$ B
A $\subset$ B $\Leftrightarrow$ A $\cap$ B = A
Question 5: Show that if A $\subset$B, then C – B $\subset$ C – A.
Answer:
Given , A $\subset$ B
To prove: C – B $\subset$ C – A
Let, x $\in$ C - B means x$\in$ C but x$\notin$B
A $\subset$ B so x$\in$ C but x$\notin$A i.e. x $\in$ C - A
Hence, C – B $\subset$ C – A
Question 6: Show that for any sets A and B,
A = ( A $\cap$ B ) $\cup$ ( A – B ) and A $\cup$ ( B – A ) = ( A $\cup$ B )
Answer:
A = ( A $\cap$ B ) $\cup$ ( A – B )
L.H.S = A = Red coloured area
R.H.S = ( A $\cap$ B ) $\cup$ ( A – B )
( A $\cap$ B ) = green coloured
( A – B ) = yellow coloured
( A $\cap$ B ) $\cup$ ( A – B ) = coloured part
Hence, L.H.S = R.H.S = Coloured part
A $\cup$ ( B – A ) = ( A $\cup$ B )
A = sky blue coloured
( B – A )=pink coloured
L.H.S = A $\cup$ ( B – A ) = sky blue coloured + pink coloured
R.H.S = ( A $\cup$ B ) = brown coloured part
L.H.S = R.H.S = Coloured part
Question 7: (i) Using properties of sets, show that
A $\cup$ ( A $\cap$ B ) = A
Answer:
(i) A $\cup$ ( A $\cap$ B ) = A
We know that A $\subset$ A
and A $\cap$ B $\subset$ A
A $\cup$ ( A $\cap$ B ) $\subset$ A
and also , A $\subset$ A $\cup$ ( A $\cap$ B )
Hence, A $\cup$ ( A $\cap$ B ) = A
Question 7: (ii) Using properties of sets, show that
A $\cap$ ( A $\cup$ B ) = A
Answer:
This can be solved as follows
(ii) A $\cap$ ( A $\cup$ B ) = A
A $\cap$ ( A $\cup$ B ) = (A $\cap$ A) $\cup$ ( A $\cap$ B )
A $\cap$ ( A $\cup$ B ) = A $\cup$ ( A $\cap$ B ) { A $\cup$ ( A $\cap$ B ) = A proved in 7(i)}
A $\cap$ ( A $\cup$ B ) = A
Question 8: Show that A $\cap$ B = A $\cap$ C need not imply B = C.
Answer:
Let, A = {0,1,2}
B = {1,2,3}
C = {1,2,3,4,5}
Given, A $\cap$ B = A $\cap$ C
L.H.S : A $\cap$ B = {1,2}
R.H.S : A $\cap$ C = {1,2}
and here {1,2,3} $\not =$ {1,2,3,4,5} = B $\not =$ C.
Hence, A $\cap$ B = A $\cap$ C need not imply B = C.
Answer:
Given, A $\cap$ X $=$B $\cap$ X $=$ $\phi$ and A $\cup$ X $=$ B $\cup$ X
To prove: A = B
A = A $\cap$(A$\cup$X) (A $\cap$ X $=$B $\cap$ X)
= A $\cap$(B$\cup$X)
= (A$\cap$B) $\cup$ (A$\cap$X)
= (A$\cap$B) $\cup$ $\phi$ (A $\cap$ X $=$ $\phi$)
= (A$\cap$B)
B = B $\cap$(B$\cup$X) (A $\cap$ X $=$B $\cap$ X)
= B $\cap$(A$\cup$X)
= (B$\cap$A) $\cup$ (B$\cap$X)
= (B$\cap$A) $\cup$ $\phi$ (B $\cap$ X $=$ $\phi$)
= (B$\cap$A)
We know that (A$\cap$B) = (B$\cap$A) = A = B
Hence, A = B
Answer:
Given, A $\cap$ B, B $\cap$ C and A $\cap$ C are non-empty sets
To prove : A $\cap$ B $\cap$ C $=$ $\phi$
Let A = {1,2}
B = {1,3}
C = {3,2}
Here, A $\cap$ B = {1}
B $\cap$ C = {3}
A $\cap$ C = {2}
and A $\cap$ B $\cap$ C $=$ $\phi$
Also read
1. Sets and set notation- A well-defined group of unique items is known as a set. They are expressed as $A=\{1,2,3\}$.
2. Venn diagrams- Venn diagrams are graphical representations of the relationships between various sets. They are drawn to understand union, intersection and complement.
3. Union and intersection of sets- The union of sets $A$ and $B$ is a set of all elements present in either $A$ or $B$ (or both). Only the components that are shared by both sets are in the intersection.
4. Set difference- The difference of two sets $A-B$ is the set of elements that are in $A$ but not in $B$.
5. Disjoint sets
When two sets have no elements in common, they are said to be disjoint; this indicates that the empty set is where they meet.
Also read
Students can also access the NCERT solutions for other subjects and make their learning feasible.
Use the links provided in the table below to get your hands on the NCERT exemplar solutions available for all the subjects.
Frequently Asked Questions (FAQs)
There are a total of seven exercises including miscellaneous exercise in the CBSE Class 11 Maths chapter 1.
Venn's diagram is useful in solving problems related to the union of sets, the intersection of sets. It is also useful in the chapters like relations and functions, probability.
φ, {–1}, {0}, {–1, 0} are the subsets of the given set.
Power set of set A = { φ, {1}, {0}, {1, 0} }
As most of the questions in the CBSE exam are not asked from the miscellaneous exercise, it is not as important for CBSE exams but it is very important for the competitive exams.
There is a total of 16 questions and 7 solved examples given in the miscellaneous exercise sets.
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