Set (Questions) ~ Mathematics Education Advocacay Nepal

Study the given Venn diagram and answer the following questions.
दिइएको भेन चित्र अध्ययन गर्नुहोस् र निम्न प्रश्नहरूको उत्तर दिनुहोस्।

समूहहरु \(R\) र \(S\) को सदस्यहरुलाई सूचिकरण गर्नुहोस ।
(List the elements of sets \(R\) and \(S\).)[1]
समूह \(R\) बाट बन्ने अनुपयुक्त उपसमूहरु लेख्नुहोस ।
(Write the improper subset formed from the set \(R\).)[1]
कुन अवस्थामा दिएको समूहहरु \(R\) र \(S\) अलगिएका समुह बन्छन् ?
(In which condition, the given sets \(R\) and \(S\) become disjoint?)[1]

From the Venn diagram, we get that
\( R = \{a, b, c\} \)
\( S = \{c, d, e\} \)
The improper subset formed from set \( R \) is
\( \{a, b, c\} \)
Sets \( R \) and \( S \) become disjoint if the element \( c \) is removed from either \( R \) or \( S \) (or both).

दिएको भेनचित्र अवलोकन गर्नुहोस। Observe the Given Venn Diagram

समूह \(A\) र \(B\) को प्रतिच्छेदन समुहमा कुन कुन सदस्यहरु पर्दछन ?
(What are the elements of the intersection of set \(A\) and set \(B\)?)[1]
समूह \(A = \{a, b, c, e\}\) बाट बन्ने कुनै दुईवटा उपयुक्त उपसमूहहरु लेख्नुहोस ।[1]
(Write any two proper subsets which can be made from set \(A = \{a, b, c, e\}\).)
समूह \(A\) र \(B\) खप्टिएका समुह हुन। यिनलाई अलगिएका समुहहरु बनाउन के गर्नुपर्छ। नयाँ बनेका अलगिएका समुहहरु लेख्नुहोस ।
(Set \(A\) and set \(B\) are overlapping sets. What should be done to make them disjoint sets? Write the newly formed disjoint sets.)[1]

From the Venn diagram, we get that
\( A \cap B = \{a, b\} \)
Any two proper subsets of \( A = \{a, b, c, e\} \) are
\( \{a, b\} \)
\( \{c, e\} \)
To make sets \( A \) and \( B \) disjoint, we must remove the common elements \( a \) and \( b \) from either \( A \) or \( B \) (or both).
One possible way is:
New \( A = \{c, e\} \)
New \( B = \{d, f\} \)

Study the given Venn diagram and answer the following questions
दिइएको भेन चित्र अध्ययन गर्नुहोस् र निम्न प्रश्नहरूको उत्तर दिनुहोस्।

अनुपयुक्त उपसमूहलाई परिभाषा गर्नुहोस ।
(Define improper subset.)[1]
समूह \(B\) को कति ओटा उपसमूह बनाउन सकिन्छ ? सुत्र प्रयोग गरी पत्ता लगाउनुहोस ।
(How many subsets of set \(B\) can be made? Find using the formula.)[1]
समूह \(A\) र \(B\) खप्टिएका वा अलगिएका समूहहरु हुन् ? कारण सहित लेख्नुहोस ।
(What type of sets are \(A\) and \(B\): overlapping or disjoint sets? Write with reason.)[1]

An improper subset of a set is the set itself.
From the Venn diagram, set \( B = \{1, 3, 5, 7, 9\} \), so it has 5 elements.
The number of subsets of a set with \( n \) elements is \( 2^n \).
Number of subsets of \( B = 2^5 = 32 \)
Sets \( A \) and \( B \) are overlapping sets.
Because they have common elements \( \{1, 3\} \), i.e., \( A \cap B \neq \emptyset \).

समूह \(M\) र \(N\) का सदस्यहरू छेउको भेन चित्रमा देखाइएको छ।
The elements of sets \(M\) and \(N\) are shown in the Venn diagram alongside.

समूह \(M\) र \(N\) का सबै सम्भावित उचित (proper) र अनुचित (improper) उपसमूहहरू लेख्नुहोस्।
Write all possible proper and improper subsets of sets \(M\) and \(N\).[1]
दिइएका समूह \(M\) र \(N\) का उपसमूहहरूको संख्या बराबर छ वा छैन, लेख्नुहोस?
Are the subsets of the given sets \(M\) and \(N\) equal in number?[1]
समूह \(M\) को कुन सदस्य हटाउँदा समुहहरु \(M\) र \(N\) अलगिएका समुह बन्छन्?
Which member of set \(M\) must be removed to make \(M\) and \(N\) disjoint sets?[1]

From the Venn diagram:
\( M = \{a, b, c\} \)
\( N = \{b, d, f\} \)

Subsets of \( M \) are

Number of Elements	Subsets	Count
0	∅	1
1	{a}, {b}, {c}	3
2	{a, b}, {a, c}, {b, c}	3
3	{a, b, c}	1
Total Subsets		8

Subsets of \( N \) are

Number of Elements	Subsets	Count
0	∅	1
1	{b}, {d}, {f}	3
2	{b, d}, {b, f}, {d, f}	3
3	{b, d, f}	1
Total Subsets		8

Yes, the number of subsets of sets \( M \) and \( N \) are equal.
Both sets have 3 elements, so each has \( 2^3 = 8 \) subsets.
To make \( M \) and \( N \) disjoint, the common element \( b \) must be removed from set \( M \).
Removing \( b \) from \( M \) gives \( M = \{a, c\} \), and then \( M \cap N = \emptyset \).

प्रश्नमा (Given)
\(P = \{\text{Prime numbers less than } 10\}\)
\(Q = \{\text{Odd numbers less than } 8\}\)
\(R = \{\text{Prime factors of } 6\}\)
\(S = \{\text{Cube numbers between } 10 \text{ and } 20\}\)

माथिका चार समूहहरूमध्ये कुन-कुन समुहहरु समतुल्य (equivalent) समूह हुन्?
Which of the four sets are equivalent sets?[1]
समूह \(R\) का उपसमूहहरू लेख्नुहोस्।
Write the subsets of set \(R\).[1]
समूह \(P\) को उपयुख्त उपसमूह समूह \(S\) हो वा होईन? कारणसहित लेख्नुहोस्।
Is the set \(S\) a proper subset of set \(P\)? Write with reason.[1]

First, list all sets:
\( P = \{2, 3, 5, 7\} \) → 4 elements
\( Q = \{1, 3, 5, 7\} \) → 4 elements
\( R = \{2, 3\} \) → 2 elements
\( S = \{\} = \emptyset \) → 0 elements (no cube numbers between 10 and 20)

Equivalent sets have the same number of elements.
So, \( P \) and \( Q \) are equivalent sets.
Set \( R = \{2, 3\} \)
Subsets of \( R \) are:
\( \emptyset, \{2\}, \{3\}, \{2, 3\} \)
Yes, \( S \) is a proper subset of \( P \), because \( S = \emptyset \).

एक भेन चित्र दिइएको छ। Given a Venn diagram

उपयूक्त उपसमूह भन्नाले के जनाउँछ?
Define proper subset.[1]
समुह \( L \) को अनुपयुक्त उपसमूह लेख्नुहोस ।
Write the improper subset of \( L \).[1]
यदि सदस्यहरु \( a, e , i ,o , u \) समुह \( M \) को मात्र हुन् भने, \( L \) र \( M \) कस्ता प्रकारका समूह हुन्? कारणसहित लेख्नुहोस्।
If \( a, e , i ,o , u \) are the members of set \( M \) only, then what type of set are \( L \) and \( M \)? Write with reason.[1]

A proper subset of a set is a subset that contains some, but not all, elements of the given set. In other words,
Set \( A \) is a proper subset of set \( B \) if every element of \( A \) is in \( B \), and \( A \neq B \)
From the Venn diagram, set \( L = \{a, b, c\} \).
The improper subset of \( L \) is the set L itself, which is
\( \{a, b, c\} \)
If \( a, e, i, o, u \) are members of set \( M \) only, then, \( L \) and \( M \) are disjoint sets because \( L \cap M = \emptyset \).

संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
Look at the adjoining Venn diagram and answer the following questions:

समुह \( A \) र \( B \) अलगिएका समुह हुन की खप्टिएका पहिचान गर्नुहोस्।
Identify whether the sets \( A \) and \( B \) are disjoint or overlapping.[1]
समुह \( B \) का उपयुक्त उप-समुहहरूको सूची तयार गर्नुहोस्।
Prepare the list of the proper subsets of set \( B \).[1]
दिइएका समुहहरूलाई अलगिएका समुह बनाउन के कस्तो समायोजन आवश्यक छ?
What adjustments are needed to ensure the given sets are disjoint?[1]

From the Venn diagram, sets \( A \) and \( B \) share common elements \( \{4, 5\} \).
Therefore, \( A \) and \( B \) are overlapping sets.

From the diagram, set \( B = \{4, 5, 6, 7\} \).
Proper subsets of \( B \) are all subsets except \( B \) itself. So, list of the proper subsets of set \( B \) are

Number of Elements	Subsets	Count
0	∅	1
1	{4}, {5}, {6}, {7}	4
2	{4, 5}, {4, 6}, {4, 7}, {5, 6}, {5, 7}, {6, 7}	6
3	{4, 5, 6}, {4, 5, 7}, {4, 6, 7}, {5, 6, 7}	4
Total		15

To make sets \( A \) and \( B \) disjoint, the common elements \( 4 \) and \( 5 \) must be removed from either \( A \) or \( B \) (or both).

संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
Look at the adjoining Venn diagram and answer the following questions:

समुह \( A \) र \( B \) अलगिएका समुह हुन की खप्टिएका पहिचान गर्नुहोस्।
Identify whether the sets \( A \) and \( B \) are disjoint or overlapping.[1]
समुह \( B \) का उपयुक्त उप-समुहहरूको सूची तयार गर्नुहोस्।
Prepare the list of the proper subsets of set \( A \).[1]
दिइएका समुहहरूलाई खप्टिएका समुह बनाउन के कस्तो समायोजन आवश्यक छ?
What adjustments are needed to ensure the given sets are overlaping?[1]

From the Venn diagram, the circles for sets \( A \) and \( B \) do not overlap, and there are no common elements shown between them.
Therefore, \( A \) and \( B \) are disjoint sets.

From the diagram, set \( A = \{x, y, z\} \).
Proper subsets of \( A \) are all subsets except \( A \) itself. which are

Number of Elements	Subsets	Count
0	∅	1
1	{x}, {y}, {z}	3
2	{x, y}, {x, z}, {y, z}	3
Total Subsets		7

It is needed to have some ellement in common then
Thus, \( A \cap B \ne \emptyset \)

संगैको भेन चित्रलाई हेरि तलका प्रश्नहरूको उत्तर दिनुहोस्।
Study the Venn diagram given alongside and answer the following questions:

समुह \( P \) र \( Q \) अलगिएका छन् कि खप्टिएका छन पहिचान गर्नुहोस्।
Identify whether the sets \( P \) and \( Q \) are disjoint or overlapping.[1]
समुह \( P \) र \( Q \) का साझा सदस्यहरुको समुह के हो?
What is the set of common elements in sets \( P \) and \( Q \)?[1]
समुह \( P \) बाट कुन सदस्यहरु हटाउदा समुह \( P \) र \( Q \) अलगिएका समुह हुन्छन?
Which elements from the set \( P \) need to be removed in order to make sets \( P \) and \( Q \) disjoint?[1]

From the Venn diagram, sets \( P \) and \( Q \) share the elements \( 6 \) and \( 7 \) in the overlapping region.
Therefore, \( P \) and \( Q \) are overlapping sets.
The common elements of sets \( P \) and \( Q \) are the elements in the intersection region.
\( P \cap Q = \{6, 7\} \)
To make sets \( P \) and \( Q \) disjoint, the common elements must be removed from set \( P \).
Remove \( 6 \) and \( 7 \) from \( P \).

Study the given Venn diagram and answer the following questions.

List the elements of \( A \).(1)
If there were no common members in \( A \) and \( B \), what types of sets would they be?(1)
Write the improper subset of set \( B \).(1)

From the Venn diagram, the elements of set \( A \) are:
\( A = \{1, 2, 3, 5, 7\} \)
If there were no common members in \( A \) and \( B \), then they would be disjoint sets.
From the diagram, set \( B = \{1, 5, 8, 9, 10\} \).
The improper subset of \( B \) is the set itself, which is
\( \{1, 5, 8, 9, 10\} \)

दिएको भेन चित्र अध्ययन गर्नुहोस् र निम्नलिखित प्रश्नहरूको उत्तर दिनुहोस ।
Study the given Venn diagram and answer the following questions.

समूह A का सदस्यहरू लेख्नुहोस |
List the elements of set A.(1)
समूह A र B को प्रतिच्छेदन \((A \cap B)\) का सदस्यहरू लेख्नुहोस|
List the elements of the intersection of sets A and B i.e. \((A \cap B)\).(1)
समूह A मा रहेका तर B मा नरहेका सदस्यहरू \((A-B)\) लेख्नुहोस|
List the elements that are in set A but not in set B i.e. \((A-B)\).

From the Venn diagram, the elements of set \( A \) are:
\( A = \{2, 3, 5, 7, 9\} \)
The intersection \( A \cap B \) consists of elements common to both \( A \) and \( B \).
\( A \cap B = \{2\} \)
The elements in \( A \) but not in \( B \) are those only in the left part of circle \( A \):
\( A - B = \{3, 5, 7, 9\} \)

दिएको भेनचित्र हेरी तल्ला प्रश्नहरूको उत्तर दिनुहोस|
Look at the given Venn diagram and answer the following questions.

समूह \(A\) र समूह \(B\) कस्ता प्रकारको समुहहरू हुन?
What types of sets are Set \(A\) and Set \(B\)?(1)
समूह \(B\) बाट कति ओटा उपसमुहहरू बन्छन?
How many subsets can be formed from Set \(B\)?(1)
\(A\) र \(B\) सर्वव्यापक समूह \(U\) का उपसमूहहरू भए यसलाई सांकेतिक रूपमा लेख्नुहोस |(1)
If \(A\) and \(B\) are subsets of the universal set \(U\), write this in symbolic form.

From the Venn diagram, sets \( A \) and \( B \) have common elements \( \{e, f\} \).
Therefore, \( A \) and \( B \) are overlapping sets.
From the diagram, set \( B = \{b, e, f, g, h\} \), which has 5 elements.
The number of subsets of a set with \( n \) elements is \( 2^n \).
Number of subsets of \( B = 2^5 = 32 \)
Since both \( A \) and \( B \) are contained within the universal set \( U \), we write:
\( A \subseteq U \) and \( B \subseteq U \)

दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
From the given Venn diagram answer the following questions.

उपयुक्त उपसमूह भनेको के हो ?
What is proper subset?(1)
\( P \) का सदस्य सङ्ख्या लेख्नुहोस् |
Write the number of elements of \( P \).(1)
\( Q \) बाट बन्ने सबै उपसमूहहरु बनाउनुहोस् |
Make all subsets of set \( Q \).(1)

A proper subset of a set is a subset that contains some but not all elements of the original set. In other words, set \( A \) is a proper subset of set \( B \) if every element of \( A \) is in \( B \), and \( A \neq B \).
Symbolically, \( A \subset B \) and \( A \neq B \).
From the Venn diagram, set \( P = \{a, b, c,d, e\} \).
Number of elements in \( P = 5 \)
From the diagram, set \( Q = \{c, d\} \).
All subsets of \( Q \) are:
\( \emptyset, \{c\}, \{d\}, \{c, d\} \)

दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
From the given Venn diagram answer the following questions.

समुह A को सदस्यहरु लेख्नुहोस ?
Write the elements of set A(1)
\( A \) र \( B \) को सम्बन्ध संकेतमा लेख्नुहोस् |
Write relation between A and B in symbol.(1)
\( B \) मा नभएका सदस्यहरु लेख्नुहोस |
Write elements which are not in B(1)

From the Venn diagram, the elements of set \( A \) are:
\( A = \{1, 2, 3,4,5, 6\} \)
Sets \( B \) is subset of \( A \)
\( B \subset A \)
The elements not in set \( B \) are
\( \{1,2,5,6\} \)

दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
From the given Venn diagram answer the following questions.

समुह A को सदस्य सँख्या लेख्नुहोस ?
Write the cardinality of set A(1)
उपसमुह भनेको के हो? लेख्नुहोस् |
What is subset, write.(1)
समुह \( A \) बाट बन्ने 2 सदस्य हुने सबै उपसमुहहरु लेख्नुहोस |
Write all subsets of A containing 2 elements(1)

From the Venn diagram, set \( A = \{q,r,s, t\} \).
Cardinality of set \( A = 4 \)
A subset is a set whose every element is also an element of another set. If all elements of set \( A \) are in set \( B \), then \( A \) is a subset of \( B \), written as \( A \subseteq B \)
Set \( A = \{q,r,s, t\} \) has 4 elements, so any twwo subsets of A are
\(\{s, t\} \) and \(\{q, r\} \)

दिएको भेनचित्रबाट सोधिका प्रश्नहरूको उत्तर दिनुहोस् |
From the given Venn diagram answer the following questions.

भेनचित्रमा कुन प्रकारका समूहहरू छन् ?
Which type of sets are shown in the Venn diagram?(1)
\( A \) का सदस्य सङ्ख्या लेख्नुहोस् |
Write the number of elements of \( A \).(1)
\( B \) बाट बन्ने कुनै एक उपसमूह लेख्नुहोस् |
Make one subset of set \( B \).(1)

From the Venn diagram, sets \( A \) and \( B \) have no common elements (no overlapping region with elements).
Therefore, \( A \) and \( B \) are disjoint sets.
From the Venn diagram, set \( A = \{1, 2, 3\} \).
Number of elements of \( A = 3 \)
From the diagram, set \( B = \{5, 7, 9\} \).
One subset of \( B \) is:
\( \{5, 7\} \)

समूहहरू \(P = \{l, o, v, e\}\) र \(Q = \{h, a, t, e\}\) दिइएको छ । (Given the set \(P = \{l, o, v, e\}\) and \(Q = \{h, a, t, e\}\).)

समूह \(P\) र \(Q\) खप्टिएको वा अलगिएका कस्ता खालका समूह हुन् ? लेख्नुहोस् ।
(What types of set \(P\) and set \(Q\) are—overlapping or disjoint? Write.)[1]
\(Q\) को 2 वटा उपयुक्त उपसमूहहरू लेख्नुहोस् ।
(Write any two proper subsets of set \(Q\).)[1]
\(P\) र \(Q\) लाई अलगिएका समूह बनाउन समूह \(P\) बाट कुन सदस्य हटाउनु पर्छ ?
(What member of set \(P\) need to be removed to make set \(P\) and \(Q\) disjoint sets?)[1]

Sets \( P = \{l, o, v, e\} \) and \( Q = \{h, a, t, e\} \) both contain the common element \( e \).
Therefore, \( P \) and \( Q \) are overlapping sets.
Two proper subsets of \( Q = \{h, a, t, e\} \) are:
\( \{h, a\} \)
\( \{t\} \)
To make \( P \) and \( Q \) disjoint, the common element \( e \) must be removed from set \( P \).
Remove \( e \) from \( P \).

Answer the following questions on the basis of the Venn diagram alongside.

Write whether sets \(N\) and \(M\) are overlapping or disjoint?[1]
Write an improper subset of set \(N\).[1]
Which social media to be removed such that sets \(N\) and \(M\) are disjoint?[1]

From the Venn diagram, sets \( N \) and \( M \) share the common element "TT" in the overlapping region.
Therefore, \( N \) and \( M \) are overlapping sets.
From the diagram, set \( N = \{F, TT\} \).
The improper subset of \( N \) is the set itself:
\( \{F, TT\} \)
To make sets \( N \) and \( M \) disjoint, the common element "TT" must be removed from set \( N \) (or from \( M \)).
Remove "TT" from set \( N \).

सर्वव्यापक समूह \( U = \{a, b, c, d, e\} \) का दुई उपसमूहहरू \( D = \{a, b, c\} \) र \( E = \{a, b, c, d, e\} \) छन् ।
(Given universal set \( U = \{a, b, c, d, e\} \), two subsets are \( D = \{a, b, c\} \) and \( E = \{a, b, c, d, e\} \).)

D र E मध्ये कुन उपयुक्त र कुन अनुपयुक्त उपसमूह हुन्, लेख्नुहोस् ।
(Between D and E, which one is the proper subset and which one is the improper subset of U? Write it.)[1K]
समूह E का कति ओटा उपसमूह बन्छन् ? लेख्नुहोस् ।
(How many subsets of set E can be formed? Write it.)[1HA]
समूहहरू D र E लाई भेनचित्रमा देखाउनुहोस् र यी दुई समूह कस्ता समूह हुन्, लेख्नुहोस् ।
(Show sets D and E in a Venn diagram and write what type of sets are D and E.)[1U]

From the given sets:
\( D = \{a, b, c\} \) is a proper subset of \( U \) because \( D \subset U \) and \( D \neq U \).
\( E = \{a, b, c, d, e\} \) is an improper subset of \( U \) because \( E = U \).
Set \( E \) has 5 elements.
Number of subsets of \( E = 2^5 = 32 \)
Sets \( D \) and \( E \) in Venn-diagram

Here
Since all elements of \( D \) are in \( E \), the set \( D \) is a subset of \( E \)

A र B सर्वव्यापक समूह \( U =\) {x : x एउटा 20 भन्दा सानो प्राकृतिक सङ्ख्या हुन्} का उपसमूहहरू हुन् । यदि \( A \)= 2 का अपवर्त्यहरू} र \( B = \) रूढ सङ्ख्याहरू} छन् ।
(A and B are subsets of the universal set \( U = \{x : x \text{ is a natural number less than } 20\} \). If \( A = \{\text{multiples of } 2\} \) and \( B = \{\text{prime numbers}\} \).)

समूह A र B लाई सूचीकरण विधिमा लेख्नुहोस् र यी खण्डिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
(List the elements of sets A and B, then write whether these two sets are overlapping sets or disjoint sets.)[1A]
माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
(Show the above sets in a Venn diagram.)[1U]
समूह A का कुनै चारओटा चार सदस्यिय उपसमूहहरू बनाउनुहोस् ।
(Make four subsets of set A having four members.)[1HA]

Elements of sets:
\( A = \{2, 4, 6, 8, 10, 12, 14, 16, 18\} \)
\( B = \{2, 3, 5, 7, 11, 13, 17, 19\} \)
They share element 2 → so they are overlapping sets.
Venn diagram:
Four subsets of A with four members are
\( \{2, 4, 6, 8\} \)
\( \{4, 6, 8, 10\} \)
\( \{8, 10, 12, 14\} \)
\( \{10, 12, 14, 16\} \)

सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5, 6\} \) का उपसमूहहरू \( P = \{2, 3, 4\} \) र \( Q = \{1, 2, 4\} \) छन् ।
(P = {2, 3, 4} and Q = {1, 2, 4} are subsets of the universal set U = {1, 2, 3, 4, 5, 6}.)

समूहहरू P र Q खण्डिएका वा अलगिएका कस्ता समूह हुन्? लेख्नुहोस् ।
(What type of sets are P and Q — overlapping or disjoint?)[1K]
समूह P का दुई सदस्यिय सबै उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set P with two members.)[1A]
समूहहरू U, P र Q लाई भेनचित्रमा देखाउनुहोस् ।
(Show sets U, P and Q in a Venn diagram.)[1U]

Set \( P = \{2, 3, 4\} \), Set \( Q = \{1, 2, 4\} \)
Common elements: 2 and 4 → so they are overlapping sets.
All 2-member subsets of P are
\( \{2, 3\} \)
\( \{2, 4\} \)
\( \{3, 4\} \)
Venn diagram is as below

दिइएको भेनचित्र अध्ययन गरी तल सोधिएका प्रश्नहरूको उत्तर दिनुहोस् ।
(Study the given Venn diagram and answer the questions asked below.)

सर्वव्यापक समूह U लाई सूचीकरण विधिबाट लेख्नुहोस् ।
(List the elements of the universal set U.)[1A]
M र N अलगिएका वा खण्डिएका कस्ता समूह हुन्, कारणसहित लेख्नुहोस् ।
(What type of sets are M and N — overlapping or disjoint sets? Write with reason.)[1A]
समूह M का कति ओटा उपसमूह बनाउन सकिन्छ? लेख्नुहोस् ।
(How many subsets of set M can be formed? Write it.)[1HA]

From the Venn diagram, the universal set \( U \) contains all elements shown inside and outside the circles.
\( U = \{3, 6, 9, 12, 15, 18, 21, 24, 27, 30\} \)
Sets \( M \) and \( N \) share common elements \( 6 \) and \( 12 \) in the overlapping region.
Therefore, \( M \) and \( N \) are overlapping sets.
Set \( M = \{3, 6, 9, 12, 15\} \), which has 5 elements.
Number of subsets of \( M = 2^5 = 32 \)

सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5, 6\} \) का दुईओटा उपसमूहहरू \( A = \{1, 3, 4, 5\} \) र \( B = \{2, 3, 5\} \) छन् ।
(Two subsets of the universal set \( U = \{1, 2, 3, 4, 5, 6\} \) are \( A = \{1, 3, 4, 5\} \) and \( B = \{2, 3, 5\} \).)

समूह A र B खण्डिएका वा अलगिएका कस्ता समूह हुन्? कारणसहित लेख्नुहोस् ।
(Are sets A and B overlapping or disjoint sets? Write with reason.)[1K]
समूह A का एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
(Write all the subsets of set A having single element.)[1A]
समूहहरू U, A र B लाई भेनचित्रमा देखाउनुहोस् ।
(Show sets U, A and B in a Venn diagram.)[1U]

Sets \( A = \{1, 3, 4, 5\} \) and \( B = \{2, 3, 5\} \) have common elements \( 3 \) and \( 5 \).
Since \( A \cap B = \{3, 5\} \neq \emptyset \), they are overlapping sets.
All single-element subsets of set \( A = \{1, 3, 4, 5\} \) are:
\( \{1\}, \{3\}, \{4\}, \{5\} \)
Venn diagram is as below

सर्वव्यापक समूह \( U = \{a, e, i, o, u\} \) का दुईओटा उपसमूहहरू \( R = \{a, i, o\} \) र \( S = \{i, o, u\} \) छन् ।
(Two subsets of the universal set \( U = \{a, e, i, o, u\} \) are \( R = \{a, i, o\} \) and \( S = \{i, o, u\} \).)

समूह R र S खण्डिएका वा अलगिएका कस्ता समूह हुन्? लेख्नुहोस् ।
(What type of sets are R and S — overlapping or disjoint?)[1K]
माथिका समुहहरुलाई भेनचित्रमा देखाउनुहोस् ।
(Show above sets in venn-diagram)[1A]
समूह R र S का कुनै दुईवटा साझा उपसमुहहरु लेख्नुहोस् ।
(Wtite any two subsets which are formed from both R and S)[1U]

Sets \( R = \{a, i, o\} \) and \( S = \{i, o, u\} \) share common elements \( i \) and \( o \).
Since \( R \cap S = \{i, o\} \neq \emptyset \), they are overlapping sets.
The venn-diagram is as below.
The common elements of \( R \) and \( S \) are \( \{i, o\} \). Any subsets formed from these common elements are subsets of both \( R \) and \( S \). Two such subsets are:
\( \{i\} \)
\( \{i, o\} \)

सर्वव्यापक समूह \( U =\) {x : x एउटा 8 भन्दा सानो प्राकृतिक सङ्ख्या हो} का उपसमूहहरू \( F = \{1, 2, 3\} \) र \( G = \{2, 3, 5\} \) छन् ।
(The subsets of the universal set \( U = \{x : x \text{ is a natural number less than } 8\} \) are \( F = \{1, 2, 3\} \) and \( G = \{2, 3, 5\} \).)

सर्वव्यापक समूह U लाई सूचीकरण विधिबाट लेख्नुहोस् र समूहहरू F र G कस्ता समूह हुन्, कारणसहित लेख्नुहोस् ।
(Write the universal set U in a listing method. Also write with reason that what type of sets F and G are — disjoint or overlapping sets.)[1A]
समूहहरू F र G का कति ओटा उपसमूह बनाउन सकिन्छ? लेख्नुहोस् ।
(How many subsets of set F and G can be made?)[1HA]
कुनै समूहको अनुपयुक्त उपसमूह भनेको कस्तो समूह हो ?
(What is the improper subset of a set?)[1K]

Natural numbers less than 8 are: 1, 2, 3, 4, 5, 6, 7.
So, \( U = \{1, 2, 3, 4, 5, 6, 7\} \)
Sets \( F = \{1, 2, 3\} \) and \( G = \{2, 3, 5\} \) share common elements \( 2 \) and \( 3 \).
Since \( F \cap G = \{2, 3\} \neq \emptyset \), they are overlapping sets.
Set \( F \) has 3 elements → Number of subsets = \( 2^3 = 8 \)
Set \( G \) has 3 elements → Number of subsets = \( 2^3 = 8 \)
So, 8 subsets can be formed from each of F and G.
An improper subset of a set is the set itself.
For example, the improper subset of \( \{1, 2, 3\} \) is \( \{1, 2, 3\} \).

A र B सर्वव्यापक समूह U का उपसमूहहरू हुन् । यदि \( U =\) {x : x एउटा 10 भन्दा सानो पूर्ण सङ्ख्या हो} , \( A =\) {y : y एउटा 10 भन्दा सानो विषम सङ्ख्या हो} र \( B =\) {z : z एउटा 10 भन्दा सानो रूढ सङ्ख्या हो} ।
(A and B are the subsets of the universal set U. If \( U = \{x : x \text{ is a whole number less than } 10\} \), \( A = \{y : y \text{ is an odd number less than } 10\} \) and \( B = \{z : z \text{ is a prime number less than } 10\} \).)

माथिका समूहहरूलाई सूचीकरण विधिमा लेख्नुहोस् ।
(List the elements of the above sets.)[1A]
समुह A र B खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
(What type of sets are A and B — overlapping or disjoint? Write it.)[1K]
समूह B का दुई सदस्यिय उपसमूहहरू लेख्नुहोस् ।
(Write the subsets of set B with two elements.)[1HA]

Whole numbers less than 10: \( 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 \)
Odd numbers less than 10: \( 1, 3, 5, 7, 9 \)
Prime numbers less than 10: \( 2, 3, 5, 7 \)
\( U = \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\} \)
\( A = \{1, 3, 5, 7, 9\} \)
\( B = \{2, 3, 5, 7\} \)
Sets \( A \) and \( B \) share common elements \( 3, 5, 7 \).
Since \( A \cap B = \{3, 5, 7\} \neq \emptyset \), they are overlapping sets.
Set \( B = \{2, 3, 5, 7\} \).
Some 2-element subsets of \( B \) are:
\( \{2, 3\} \)
\( \{5, 7\} \)

एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)

अनुपयुक्त उपसमूह लाई परिभाषित गर्नुहोस् । Define improper subset.[1K]
समूह B का 2 सदस्यिय उपयुक्त उपसमूहहरू लेख्नुहोस् ।
(Write proper subsets of set B with two members.)[1U]
समूहहरू A र B खण्डिएका समूहहरू हुन् । समर्थन गर्नुहोस् ।
(Sets A and B are overlapping sets. Justify it.)[1HA]

अनुपयुक्त उपसमूह लाई परिभाषित गर्नुहोस् । (Define improper subset.)[1K]
An improper subset is a subset of a set that includes all the elements of the original set, i.e., the set itself.
समूह B का 2 सदस्यिय उपयुक्त उपसमूहहरू लेख्नुहोस् ।
(Write proper subsets of set B with two members.)[1U]
The proper subsets of set S with two members are {9, 10}, {9, 12}, and {10, 12}.
समूहहरू A र B खण्डिएका समूहहरू हुन् । समर्थन गर्नुहोस् ।
(Sets A and B are overlapping sets. Justify it.)[1HA]
Sets A and B are overlapping because they have a common element, 9, as indicated by their non-empty intersection (R ∩ S = {9}).

P = {a, r, e} र Q = {a, r, t} सर्वव्यापक समूह U = {a, r, t, e, s} का उपसमूहहरू हुन् ।
(P = {a, r, e} and Q = {a, r, t} are subsets of the universal set U = {a, r, t, e, s}.)

उपयुक्त समूहलाई परिभाषित गर्नुहोस् । Define proper subset.[1K]
यो कुन समूह हो जुन सबै समूहहरूको उपसमूह हो ?
(What is the set which is subset of every set?)[1U]
यदि समूह P = {e, s} भएको भए, समूहहरू P र Q बीचको सम्बन्ध कस्तो हुन्थ्यो ? कारणसहित लेख्नुहोस् ।
(If set P = {e, s}, what will be the relation between set P and Q? Write with reason.)[1HA]

उपयुक्त समूहलाई परिभाषित गर्नुहोस् । (Define proper subset.)[1K]
A proper subset of a set is a subset that contains some, but not all, elements of the original set. In other words, if \( A \) is a proper subset of \( B \), then \( A \subseteq B \) and \( A \neq B \).
For example, \( \{a, r\} \) is a proper subset of \( \{a, r, e\} \).
यो कुन समूह हो जुन सबै समूहहरूको उपसमूह हो ?
(What is the set which is subset of every set?)[1U]
The empty set, denoted \( \emptyset \), is a subset of every set.
यदि समूह P = {e, s} भएको भए, समूहहरू P र Q बीचको सम्बन्ध कस्तो हुन्थ्यो ? कारणसहित लेख्नुहोस् ।
(If set P = {e, s}, what will be the relation between set P and Q? Write with reason.)[1HA]
If \( P = \{e, s\} \) and \( Q = \{a, r, t\} \), then (\( P \cap Q = \emptyset \)), so the sets \( P \) and \( Q \) are disjoint sets.

सर्वव्यापक समूह \( U = \{1, 2, 3, 4, 5\} \) का दुईओटा उपसमूहहरू \( D = \{1, 2, 3\} \) र \( E = \{3, 4, 5\} \) छन् ।
(Two subsets of the universal set \( U = \{1, 2, 3, 4, 5\} \) are \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \).)

खप्टिएका समूहलाई परिभाषित गर्नुहोस् । Define overlapping sets.[1K]
D र E खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
(Write common subsets of the sets D and E.)[1U]
समूह D र E बाट कुन सदस्य हटाउँदा यी समूहहरू अलगिएका समूह बन्छन् ?
(Which element of set D and set E should be removed to make them disjoint sets?)[1HA]

खप्टिएका समूहलाई परिभाषित गर्नुहोस् । (Define overlapping sets.)[1K]
Overlapping sets are sets that have at least one common element. In other words, if two sets \( A \) and \( B \) are overlapping, then their intersection is not empty, i.e., \( A \cap B \neq \emptyset \).
D र E खप्टिएका वा अलगिएका कस्ता समूह हुन्, लेख्नुहोस् ।
(Write common subsets of the sets D and E.)[1U]
The common elements of \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \) are \( D \cap E = \{3\} \). So,
D and E are overlaping sets.
समूह D र E बाट कुन सदस्य हटाउँदा यी समूहहरू अलगिएका समूह बन्छन् ?
(Which element of set D and set E should be removed to make them disjoint sets?)[1HA]
Sets \( D = \{1, 2, 3\} \) and \( E = \{3, 4, 5\} \) have a common element, \( 3 \), since \( D \cap E = \{3\} \). So, removing the element \( 3 \) from either \( D \) or \( E \) makes them disjoint sets.

सँगै दिइएको भेनचित्र अध्ययन गरी तलका प्रश्नहरूको उत्तर दिनुहोस् ।
(Study the given Venn diagram, then answer the following questions.)

कस्ता समूहहरूलाई अलगिएका समूह भनिन्छ ? लेख्नुहोस् ।
(What type of sets are disjoint sets? Write.)[1K]
समूह R का सबै एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set R with single elements.)[1U]
कुन अवस्थामा दिइएका समूहहरू R र S अलगिएका समूह बन्छन् ?
(In what condition, the given sets R and S become disjoint sets?)[1HA]

कस्ता समूहहरूलाई अलगिएका समूह भनिन्छ ? (What type of sets are disjoint sets? Write.)[1K]
Disjoint sets are sets that have no common elements. In other words, if two sets \( A \) and \( B \) are disjoint, then their intersection is empty, i.e., \( A \cap B = \emptyset \).
समूह R का सबै एक सदस्यिय उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set R with single elements.)[1U]
From the Venn diagram, set \( R = \{a, b\} \). The single-element subsets of \( R \) are:
\( \{a\}, \{b\} \).
कुन अवस्थामा दिइएका समूहहरू R र S अलगिएका समूह बन्छन् ?
(In what condition, the given sets R and S become disjoint sets?)[1HA]
From the Venn diagram, \( R = \{a, b\} \) and \( S = \{a,b,c, d\} \), with \( R \cap S = \{a, b\} \). To make \( R \) and \( S \) disjoint, R must not contain any elements from \( S = \{a,b,c, d\} \), it can contain elements from \( \{e, f\}\).

A = {x : x ≤ 3, x ∈ N} र B = {y : y ≤ 4, y ∈ N} सर्वव्यापक समूह U = {z : z ≤ 6, z ∈ W} का उपसमूहहरू हुन् ।
(A = {x : x ≤ 3, x ∈ N} and B = {y : y ≤ 4, y ∈ N} are subsets of the universal set U = {z : z ≤ 6, z ∈ W}.)

समूहहरू A, B र U लाई सूचीकरण विधिबाट लेख्नुहोस् ।
(Write the sets A, B and U in listing method.)[1A]
समूहहरू A र B का दुई सदस्यिय साझा उपसमूहहरू लेख्नुहोस् ।
(Write all the common subsets of set A and B with two elements.)[1U]
समूह B को कुन सदस्य हटाउँदा A र B एकअर्काको अनुपयुक्त उपसमूह बन्छन् ?
(Which element of set B is to be removed so that they become improper subset of A and B each other?)[1HA]

Natural numbers (N) start from 1, and whole numbers (W) include 0.
\( A = \{x : x \leq 3, x \in \mathbb{N}\} = \{1, 2, 3\} \)
\( B = \{y : y \leq 4, y \in \mathbb{N}\} = \{1, 2, 3, 4\} \)
\( U = \{z : z \leq 6, z \in \mathbb{W}\} = \{0, 1, 2, 3, 4, 5, 6\} \)
Common elements of \( A \) and \( B \): \( A \cap B = \{1, 2, 3\} \)
All 2-element subsets of this intersection are the common 2-element subsets of both \( A \) and \( B \), which are
\( \{1, 2\} \)
\( \{1, 3\} \)
\( \{2, 3\} \)
Currently, \( A = \{1, 2, 3\} \) and \( B = \{1, 2, 3, 4\} \).
For \( A \) and \( B \) to be improper subsets of each other, the element \( 4 \) must be removed from set \( B \)

एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)

कस्तो समूहलाई दिइएको समूहको उपसमूह भनिन्छ ? लेख्नुहोस् ।
(What type of set is called subset of the given set? Write it.)[1K]
समूहहरू A, B र U बीच कस्तो सम्बन्ध छ ? लेख्नुहोस् ।
(What is the relation between sets A, B and U? Write it.)[1U]
समूहहरू A र B लाई बराबर समूह बनाउन, समूह A बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
(To make sets A and B equal, which elements of set A are to be removed?)[1HA]

कस्तो समूहलाई दिइएको समूहको उपसमूह भनिन्छ ? (What type of set is called subset of the given set? Write it.)[1K]
A set is called a subset of a given set if all its elements are also elements of the given set. In other words, if every element of set \( A \) is contained in set \( B \), then \( A \subseteq B \)
समूहहरू A, B र U बीच कस्तो सम्बन्ध छ ? लेख्नुहोस् ।
(What is the relation between sets A, B and U? Write it.)[1U]
From the Venn diagram, \( U = \{a, b, c, d, e, f\} \), \( A = \{a, b, c, d\} \), and \( B = \{a, b\} \). Since all elements of \( A \) and \( B \) are in \( U \), and all elements of \( B \) are in \( A \), the relations are follows.
\( B \subseteq A \) and \( A \subseteq U \), \( B \subseteq U \).
समूहहरू A र B लाई बराबर समूह बनाउन, समूह A बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
(To make sets A and B equal, which elements of set A are to be removed?)[1HA]
From the Venn diagram, \( A = \{a, b, c, d\} \) and \( B = \{a, b\} \). To make \( A \) equal to \( B \), the elements \( c \) and \( d \) must be removed from \( A \)

एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)

P र Q खण्डिएका वा अलगिएका कस्ता समूह हुन् ? कारणसहित लेख्नुहोस् ।
(What types of sets are the sets P and Q — overlapping or disjoint sets? Write with reason.)[1K]
समूह P का सबै उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set P.)[1U]
समूहहरू P र Q समूह U का कस्ता उपसमूहहरू हुन्, कारणसहित लेख्नुहोस् ।
(What type of subsets of set U are the sets P and Q? Write with reason.)[1K]

From the Venn diagram, set \( P = \{x, y\} \) and set \( Q = \{w, z\} \).
There is no overlapping region with common elements between \( P \) and \( Q \).
Since \( P \cap Q = \emptyset \), sets \( P \) and \( Q \) are disjoint sets.
Set \( P = \{x, y\} \) has 2 elements.
All subsets of \( P \) are:
\( \emptyset, \{x\}, \{y\}, \{x, y\} \)
From the diagram:
Universal set \( U = \{x, y, w, z, u, v\} \)
\( P = \{x, y\} \subset U \) and \( Q = \{w, z\} \subset U \)
Therefore, both \( P \) and \( Q \) are proper subsets of \( U \), because all their elements are in \( U \) and neither equals \( U \).

सर्वव्यापक समूह \( U = \) {6 भन्दा साना प्राकृतिक सङ्ख्याहरू} का दुईओटा उपसमूहहरू \( A =\) {6 भन्दा साना जोर सङ्ख्याहरू} र \( B =\) {6 भन्दा साना रूढ सङ्ख्याहरू} छन् ।
(Two subsets of the universal set \( U = \{\text{Natural numbers less than } 6\} \) are \( A = \{\text{Even numbers less than } 6\} \) and \( B = \{\text{Prime numbers less than } 6\} \).)

समूहहरू U, A र B लाई सूचीकरण विधिमा लेख्नुहोस् ।
(Write the sets U, A and B in listing method.)[1A]
माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
(Represent the above sets in Venn diagram.)[1U]
समूह A का सबै उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set A.)[1U]

समूहहरू U, A र B लाई सूचीकरण विधिमा लेख्नुहोस् । (Write the sets U, A and B in listing method.)[1A]
The sets are
\( U = \{1, 2, 3, 4, 5\} \)
\( A = \{2, 4\} \)
\( B = \{2, 3, 5\} \)
माथिका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
(Represent the above sets in Venn diagram.)[1U]
The venn-diagram is
समूह A का सबै उपसमूहहरू लेख्नुहोस् ।
(Write all subsets of set A.)[1U]
Set \( A = \{2, 4\} \) has 2 elements, so it has \( 2^2 = 4 \) subsets.
So, the subsets are
\( \emptyset, \{2\}, \{4\}, \{2, 4\} \).

एउटा भेनचित्र दिइएको छ । (A Venn diagram is given.)

कुनै समूह र यसको अनुपयुक्त उपसमूहको सम्बन्ध कति हुन्छ ?
(What is the relation of a set to its improper subset?)[1K]
समूह U का जम्मा कति ओटा उपसमूह निर्माण गर्न सकिन्छ ?
(How many subsets of set U can be constructed?)[1U]
समूहहरू R र S लाई अलगिएका समूह बनाउन समूह S बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
(Which elements of set S are to be removed to make the sets R and S disjoint sets?)[1HA]

कुनै समूह र यसको अनुपयुक्त उपसमूहको सम्बन्ध कति हुन्छ ? (What is the relation of a set to its improper subset?)[1K]
An improper subset of a set is the set itself. Thus, the relation of a set to its improper subset is equality, i.e., if \( A \) is a set, its improper subset is \( A \), so \( A = A \).
समूह U का जम्मा कति ओटा उपसमूह निर्माण गर्न सकिन्छ ?
(How many subsets of set U can be constructed?)[1U]
From the Venn diagram, \( U = \{a, b, c, d, e, f, g\} \) has 7 elements. The number of subsets of a set with \( n \) elements is \( 2^n \).
Thus, \( 2^7 = 128 \) subsets can be constructed from set \( U \).
समूहहरू R र S लाई अलगिएका समूह बनाउन समूह S बाट कुन कुन सदस्यहरू हटाउनुपर्छ ?
(Which elements of set S are to be removed to make the sets R and S disjoint sets?)[1HA]
From the Venn diagram, \( R = \{a, b, c\} \), \( S = \{b, c, d\} \), and their intersection is \( R \cap S = \{b, c\} \). To make \( R \) and \( S \) disjoint, \( b \) and \( c \) must be removed from \( S \).

सर्वव्यापक समूह \( U =\) {7 भन्दा साना पूर्ण सङ्ख्याहरू} का दुईओटा उपसमूहहरू \( A =\) {x : x एउटा विषम सङ्ख्या हो} र \( B =\) {y : y एउटा रूढ सङ्ख्या हो} छन् ।
(Two subsets of the universal set \( U = \{\text{whole numbers less than } 7\} \) are \( A = \{x : x \text{ is an odd number}\} \) and \( B = \{y : y \text{ is a prime number}\} \).)

उपसमूहलाई परिभाषित गर्नुहोस् । Define subset.[1K]
माथि दिइएका समूहहरूलाई भेनचित्रमा देखाउनुहोस् ।
(Show the above sets in Venn diagram.)[1U]
समूहहरू A र B खण्डिएका वा अलगिएका कस्ता समूह हुन् ? कारणसहित लेख्नुहोस् ।
(What type of sets are A and B — disjoint or overlapping? Write with reason.)[1HA]

A set \( X \) is called a subset of another set \( Y \) if every element of \( X \) is also an element of \( Y \).
It is denoted by \( X \subseteq Y \).
Here
Whole numbers less than 7: \( U = \{0, 1, 2, 3, 4, 5, 6\} \)
Odd numbers: \( A = \{1, 3, 5\} \)
Prime numbers: \( B = \{2, 3, 5\} \)
Common elements: \( 3 \) and \( 5 \)
The venn diagram is as below.
Since \( A \cap B = \{3, 5\} \neq \emptyset \), sets \( A \) and \( B \) are overlapping sets.

Set (Questions)

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