JEE Mathematics Set Operations Relations and Mappings MCQs

Type – 1

Question. Let A and B be two sets such that \( A \cup B = A \). Then \( A \cap B \) is equal to
(a) \( \phi \)
(b) B
(c) A
(d) None of the options
Answer: (b) B

Question. Let A and B be two sets. Then \( (A \cup B)' \cup (A' \cap B) \) is equal to
(a) \( A' \)
(b) A
(c) \( B' \)
(d) None of the options
Answer: (a) \( A' \)

Question. Let U be the universal set and \( A \cup B \cup C \subseteq U \). Then \( \{(A-B) \cup (B - C) \cup (C-A)\}' \) is equal to
(a) \( A \cup B \cup C \)
(b) \( A \cup (B \cup C) \)
(c) \( A \cap B \cap C \)
(d) \( A \cap (B \cup C) \)
Answer: (c) \( A \cap B \cap C \)

Question. If A and B are two sets then \( (A - B) \cup (B - A) \cup (A \cap B) \) is equal to
(a) \( A \cup B \)
(b) \( A \cap B \)
(c) A
(d) \( B' \)
Answer: (a) \( A \cup B \)

Question. 20 teachers of a school either teach mathematics or physics. 12 of them teach mathematics while 4 teach both the subjects. Then the number of teachers teaching only physics is
(a) 12
(b) 8
(c) 16
(d) None of the options
Answer: (b) 8

Question. Of the members of three athletic teams in a school, 21 are in the cricket team, 26 are in the hockey team and 29 are in the football team. Among them, 14 play hockey and cricket, 15 play hockey and football, and 12 play football and cricket. Eight play all the three games. The total number of members in the three athletic teams is
(a) 43
(b) 76
(c) 49
(d) None of the options
Answer: (a) 43

Question. The relation "congruence modulo m" is
(a) reflexive only
(b) transitive only
(c) symmetric only
(d) an equivalence relation
Answer: (d) an equivalence relation

Question. R is a relation over the set of real numbers and it is given by \( mn \ge 0 \). Then R is
(a) symmetric and transitive
(b) reflexive and symmetric
(c) a partial-order relation
(d) an equivalence relation
Answer: (d) an equivalence relation

Question. R is a relation over the set of integers and it is given by \( (x, y) \in R \iff |x - y| \le 1 \). Then R is
(a) reflexive and transitive
(b) reflexive and symmetric
(c) symmetric and transitive
(d) an equivalence relation
Answer: (b) reflexive and symmetric

Question. Let r be a relation over the set \( N \times N \) and it is defined by \( (a, b) r (c, d) \Rightarrow a + d = b + c \). Then r is
(a) reflexive only
(b) symmetric only
(c) transitive only
(d) an equivalence relation
Answer: (d) an equivalence relation

Question. Let \( A = \{1,2,3\} \). The total number of distinct relations that can be defined
(a) \( 2^9 \)
(b) 6
(c) 8
(d) None of the options
Answer: (a) \( 2^9 \)

Question. On the set \( A = \{1, 2, 3, 4\} \), a relation is \( R = \{(1, 3), (4, 2), (2, 4), (2, 3), (3, 1)\} \). The relation R is
(a) a function
(b) transitive
(c) not symmetric
(d) reflexive
Answer: (c) not symmetric

Question. Let \( R = \{(3, 3), (6, 6), (9, 9), (12, 12), (6, 12), (3, 9), (3, 12), (3, 6)\} \) be a relation on the set \( A = \{3, 6, 9, 12\} \). The relation is
(a) reflexive only
(b) reflexive and transitive only
(c) reflexive and symmetric only
(d) an equivalence relation
Answer: (a) reflexive only

Question. The range of the function \( f(x) = {}^{7-x}P_{x-3} \) is where \( {}^n P_r = \frac{n!}{(n-r)!} \)
(a) {1,2,3}
(b) {1,2,3,4,5,6}
(c) {1,2,3,4}
(d) {1,3}
Answer: (a) {1,2,3}

Question. Let \( R \) = set of real numbers and \( R_c \) = set of real angles in radian measure. If \( f: R_c \to R \) be a mapping such that \( f(x) = \sin x, x \in R_c \), then f is
(a) one-one and into
(b) one-one and onto
(c) many-one and onto
(d) many-one and into
Answer: (d) many-one and into

Question. Let \( f : R \to R \) such that \( f(x) = \frac{1}{1 + x^2}, x \in R \). Then f is
(a) injective
(b) surjective
(c) bijective
(d) None of the options
Answer: (d) None of the options

Question. \( f: R \times R \to R \) such that \( f(x + iy) = \sqrt{x^2 + y^2} \). Then f is
(a) many-one and into
(b) one-one and onto
(c) many-one and onto
(d) one-one and into
Answer: (a) many-one and into

Question. Let \( A = \{x| - 1 \le x \le 1\} \). If \( f: A \to A \) be bijective then a possible definition of \( f(x) \) is
(a) \( |x| \)
(b) \( x |x| \)
(c) \( \sin \pi x \)
(d) None of the options
Answer: (d) None of the options

Question. Let \( A = \{1, 2, 3\} \) and \( B = \{a, b\} \). Which of the following subsets of \( A \times B \) is a mapping from A to B?
(a) \( \{(1, a), (3, b), (2, a), (2, b)\} \)
(b) \( \{(1, b), (2, a), (3, a)\} \)
(c) \( \{(1, a), (2, b)\} \)
(d) None of the options
Answer: (b) \( \{(1, b), (2, a), (3, a)\} \)

Type 2

Choose the correct options. One or more options may be correct.

Question. Let R be the relation over the set of straight lines of a plane such that \( l_1 R l_2 \iff l_1 \perp l_2 \). Then, R is
(a) symmetric
(b) reflexive
(c) transitive
(d) an equivalence relation
Answer: (a) symmetric

Question. Let R be the relation over the set of integers such that \( m R n \) if and only if m is a multiple of n. Then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) an equivalence relation
Answer: (a) reflexive, (c) transitive

Question. Let \( A = \{1, 2, 3, 4\} \) and R be a relation in A given by \( R = \{(1, 1), (2, 2), (3, 3), (4, 4), (1, 2), (2, 1), (3, 1), (1, 3)\} \). Then R is
(a) reflexive
(b) symmetric
(c) transitive
(d) an equivalence relation
Answer: (a) reflexive, (b) symmetric

Question. Let \( f: R \to R \) be a mapping such that \( f(x) = \frac{x^2}{1 + x^2} \). Then f is
(a) many–one
(b) one–one
(c) into
(d) onto
Answer: (a) many–one, (c) into

Question. Let \( A = \{1, 2, 3\} \) and \( B = \{a, b, c\} \). If f is a function from A to B and g is a one-one function from A to B then the maximum number of definitions
(a) f is 9
(b) g is 9
(c) f is 27
(d) g is 6
Answer: (c) f is 27, (d) g is 6