SET & RELATIONS
| Site: | eduvertex |
| Course: | MATHS FOR JEE |
| Book: | SET & RELATIONS |
| Printed by: | Guest user |
| Date: | Sunday, 19 May 2024, 2:19 AM |
1. DPP-1
\title{
Topic :- SETS
}
1. Let \(R_{1}\) be a relation defined by \(R_{1}=\{(a, b) \mid a \geq b, a, b \in R\}\). Then, \(R_{1}\) is
a) An equivalence relation on \(R\)
b) Reflexive, transitive but not symmetric
c) Symmetric, transitive but not reflexive
d) Neither transitive not reflexive but symmetric
2. On the set of human beings a relation \(R\) is defined as follows:
"aRb iff \(a\) and \(b\) have the same brother". Then \(R\) is
a) Only reflexive
b) Only symmetric
c) Only transitive
d) Equivalence
3. In a class of 35 students, 17 have taken Mathematics, 10 have taken Mathematics but not Economics. If each student has taken either Mathematics or Economics or both, then the number of students who have taken Economics but not Mathematics is
a) 7
b) 25
c) 18
d) 32
4. \(\quad\{n(n+1)(2 n+1): n \in Z\} \subset\)
a) \(\{6 k: k \in Z\}\)
b) \(\{12 k: k \in Z\}\)
c) \(\{18 k: k \in Z\}\)
d) \(\{24 k: k \in Z\}\)
5. If \(A=\{1,2,3,4,5\}, B=\{2,4,6\}, C=\{3,4,6\}\), then \((A \cup B) \cap C\) is
a) \(\{3,4,6\}\)
b) \(\{1,2,3\}\)
c) \(\{1,4,3\}\)
d) None of these
6. Let \(A\) be the set of all students in a school. A relation \(R\) is defined on \(A\) as follows: "aRb iff \(a\) and \(b\) have the same teacher"
a) Reflexive
b) Symmetric
c) Transitive
d) Equivalence
7. If \(P\) is the set of all parallelograms, and \(T\) is the set of all trapeziums, then \(P \cap T\) is
a) \(P\)
b) \(T\)
c) \(\phi\)
d) None of these
8. \(A\) and \(B\) are any two non-empty sets and \(A\) is proper subset of \(B\). If \(n(A)=5\), then find the minimum possible value of \(n(A \Delta B)\)
a) Is 1
b) Is 5
c) Cannot be determined
d) None of these
9. If \(n(A)=4, n(B)=3, n(A \times B \times C)=240\), then \(n(C)\) is equal to
a) 288
b) 1
c) 12
d) 2
10. In a class, 70 students wrote two tests viz; test-I and test-II. \(50 \%\) of the students failed in test-I and \(40 \%\) of the students in test-II. How many students passed in both tests?
a) 21
b) 7
c) 28
d) 14
11. Let \(Z\) denote the set of all integers and \(A=\left\{(a, b): a^{2}+3 b^{2}=28, \quad a, b \in Z\right\}\) and \(B=\{(a, b)\) \(: a>b, a, b \in Z\}\). Then, the number of elements in \(A \cap B\) is
a) 2
b) 3
c) 4
d) 6 12. Let \(L\) be the set of all straight lines in the Euclidean plane. Two lines \(l_{1}\) and \(l_{2}\) are said to be related by the relation \(R\) iff \(l_{1}\) is parallel to \(l_{2}\). Then, the relation \(R\) is not
a) Reflexive
b) Symmetric
c) Transitive
d) None of these
13. Let \(R\) be a relation on the set \(N\) be defined by \(\{(x, y) \mid x, y \in N, 2 x+y=41\}\). Then, \(R\) is
a) Reflexive
b) Symmetric
c) Transitive
d) None of these
14. In an office, every employee likes at least one of tea, coffee and milk. The number of employees who like only tea, only coffee, only milk and all the three are all equal. The number of employees who like only tea and coffee, only coffee and milk and only tea and milk are equal and each is equal to the number of employees who like all the three. Then a possible value of the number of employees in the office is
a) 65
b) 90
c) 77
d) 85
15. Which of the following cannot be the number of elements in the power set of any finite set?
a) 26
b) 32
c) 8
d) 16
16. The relation 'is subset of' on the power set \(P(A)\) of a set \(A\) is
a) Symmetric
b) Anti-symmetric
c) Equivalence relation
d) None of these
17. Let \(A\) and \(B\) be two non-empty subsets of a set \(X\) such that \(A\) is not a subset of \(B\). Then,
a) \(A\) is a subset of complement of \(B\)
b) \(B\) is a subset of \(A\)
c) \(A\) and \(\mathrm{B}\) are disjoint
d) \(A\) and the complement of \(B\) are non-disjoint
18. If \(A, B\) and \(C\) are three sets such that \(A \supset B \supset C\), then \((A \cup B \cup C)-(A \cap B \cap C)=\)
a) \(A-B\)
b) \(B-C\)
c) \(A-C\)
d) None of these
19. A survey shows that \(63 \%\) of the Americans like cheese whereas \(76 \%\) like apples. If \(x \%\) of the Americans like both cheese and apples, then
a) \(x=39\)
b) \(x=63\)
c) \(39 \leq x \leq 63\)
d) None of these
20. If \(X=\left\{4^{n}-3 n-1: n \in N\right\}\) and \(Y=\{9(n-1): n \in N\}\), then \(X \cup Y\) is equal to
a) \(X\)
b) \(Y\)
c) \(N\)
d) None of these
2. DPP-2
\title{
Topic :- SETS
}
1. Let \(A=\{x: x\) is a multiple of 3\(\}\) and \(B=\{x: x\) is a multiple of 5\(\}\). Then, \(A \cap B\) is given by
a) \(\{3,6,9, \ldots \ldots\}\)
b) \(\{5,10,15,20, \ldots \ldots\).
c) \(\{15,30,45, \ldots . .\).
d) None of these
2. If \(n(A \times B)=45\), then \(n(A)\) cannot be
a) 15
b) 17
c) 5
d) 9
3. In order that a relation \(R\) defined on a non-empty set \(A\) is an equivalence relation, it is sufficient, if \(R\)
a) Is reflective
b) Is symmetric
c) Is transitive
d) Possesses all the above three properties
4. For real numbers \(x\) and \(y\), we write \(x R y \Leftrightarrow x-y+\sqrt{2}\) is an irrational number. Then, the relation \(R\) is
a) Reflexive
b) Symmetric
c) Transitive
d) None of these
5. In a class of 45 students, 22 can speak Hindi and 12 can speak English only. The number of students, who can speak both Hindi and English, is
a) 9
b) 11
c) 23
d) 17
6. \(\quad A, B\) and \(C\) are three non-empty sets. If \(A \subset B\) and \(B \subset C\), then which of the following is true?
a) \(B-A=C-B\)
b) \(A \cap B \cap C=B\)
c) \(A \cup B=B \cap C\)
d) \(A \cup B \cup C=A\)
7. \(\left\{x \in R: \frac{2 x-1}{x^{3}+4 x^{2}+3 x} \in R\right\}\) equals
a) \(R-\{0\}\)
b) \(R-\{0,1,3\}\)
c) \(R-\{0,-1,-3\}\)
d) \(R-\) \(\left\{0,-1,-3,+\frac{1}{2}\right\}\)
8. If \(R\) is a relation from a finite set \(A\) having \(m\) elements to a finite \(\operatorname{set} B\) having \(n\) elements, then the number of relations from \(A\) to \(B\) is
a) \(2^{m n}\)
b) \(2^{m n}-1\)
c) \(2 m n\)
d) \(m^{n}\)
9. If \(A=\left\{(x, y): y^{2}=x ; x, y \in R\right\}\) and
\(B=\{(x, y): y=|x| ; x, y \in R\}\), then
a) \(A \cap B=\phi\)
b) \(A \cap B\) is a singleton set
c) \(A \cap B\) contains two elements only
d) \(A \cap B\) contains three elements only
10. Which of the following is an equivalence relation?
a) Is father of
b) Is less than
c) Is congruent to
d) Is an uncle of
11. From 50 students taking examinations in Mathematics, Physics and Chemistry, 37 passed Mathematics, 24 Physics and 43 Chemistry. At most 19 passed Mathematics and Physics, at most 29 passed Mathematics and Chemistry and at most 20 passed Physics and Chemistry. The largest possible number that could have passed all three examinations is
a) 11
b) 12
c) 13
d) 14
12. Let \(A\) be the non-void set of the children in a family. The relation ' \(x\) is a brother of \(y^{\prime}\) on \(A\) is
a) Reflexive
b) Symmetric
c) Transitive
d) None of these
13. In a class of 30 pupils 12 take needls work, 16 take physics and 18 take history. If all the 30 students take at least one subject and no one takes all three, then the number of pupils taking 2 subjects is
a) 16
b) 6
c) 8
d) 20
14. If \(R\) is a relation on a finite set having \(n\) elements, then the number of relations on \(A\) is
a) \(2^{n}\)
b) \(2^{n^{2}}\)
c) \(n^{2}\)
d) \(n^{n}\)
15. The void relation on a set \(A\) is
a) Reflexive
b) Symmetric and transitive
c) Reflexive and symmetric
d) Reflexive and transitive
16. Suppose \(A_{1}, A_{2}, \ldots, A_{30}\) are thirty sets, each having 5 elements and \(B_{1}, B_{2}, \ldots, B_{n}\) are \(n\) sets each with 3 elements, let \(\bigcup_{i=1}^{30} A_{i}=\bigcup_{j=1}^{n} B_{j}=S\) and each element of \(S\) belongs to exactly 10 of the \(A_{i}\) 's and exactly 9 of the \(B_{j}\) 's. Then, \(n\) is equal to
a) 115
b) 83
c) 45
d) None of these
117.If \(A\) is a finite set having \(n\) elements, then \(P(A)\) has
a) \(2 n\) elements
b) \(2^{n}\) elements
c) \(n\) elements
d) None of these
18. Let \(A\) and \(B\) have 3 and 6 elements respectively. What can be the minimum number of elements in \(A \cup B\) ?
a) 3
b) 6
c) 9
d) 18
19. Let \(R\) be a reflexive relation on a set \(A\) and \(I\) be the identity relation on \(A\). Then,
a) \(R \subset I\)
b) \(I \subset R\)
c) \(R=I\)
d) None of these
20. If \(A_{1}, A_{2}, \ldots, A_{100}\) are sets such that \(n\left(A_{i}\right)=i+2, A_{1} \subset A_{2} \subset A_{3} \ldots \subset A_{100}\) and \(\bigcap_{i=3}^{100} A_{i}=A\), then \(n(A)\)
a) 3
b) 4
c) 5
d) 6