Chapter 1

Let \(F_{1}\) be the set of all parallelograms, \(F_{2}\) the set of rectangles, \(F_{3}\) the set of rhombuses, \(F_{4}\) the set of squares and \(F_{5}\) the set of trapeziums in a plane then \(F_{1}\) is equal to (A) \(F_{2} \cap F_{3}\) (B) \(F_{2} \cup F_{3} \cup F_{4} \cup F_{1}\) (C) \(F_{3} \cap F_{4}\) (D) None of these

(i) Let \(R\) be the relation on the set \(R\) of all real numbers defined by setting \(a R b\) if \(|a-b| \leq \frac{1}{2}\). Then \(R\) is (A) Reflexive and symmetric but not transitive (B) Symmetric and transitive but not reflexive (C) Transitive but neither reflexive nor symmetric (D) None of these

\(n / m\) means that \(n\) is a factor of \(m\), then the relation \(\%\) is (A) reflexive and symmetric. (B) transitive and reflexive. (C) reflexive, transitive and symmetric. (D) reflexive, transitive and not symmetric.

Set \(A\) and \(B\) have 3 and 6 elements respectively. What can be the minimum number of elements in \(A \cup B\) ? (A) 18 (B) 9 (C) 6 (D) 3

Let \(R\) be a relation defined on the set of natural numbers \(N\) as \(R=[(x, y): x \in N, y \in N, 2 x+y=41]\). Then (A) Domain of \(R=\\{1,2,3, \ldots, 19,20\\}\) (B) Range of \(R=\\{39,37,35,9,7,5,3,1\\}\) (C) \(R\) is reflexive (D) \(R\) is symmetric

Let \(A=\\{x: x \in R,|x|<1\\}\) \(B=\\{x: x \in R,|x-1| \geq 1\\}\) and \(A \cup B=R-D\), then the set \(D\) is (A) \(\\{x: 1

Consider the set \(A\) of all determinants of order 3 with entries 0 or 1 only. Let \(B\) be subset of \(A\) consisting of all determinants with value \(1 .\) Let \(C\) be the subset \(A\) of consisting of all determinants with value \(-1\). Then (A) \(C\) is empty. (B) \(B\) has as many elements as \(C\). (C) \(A=B \cup C\). (D) \(B\) has twice as many elements as \(C\).

Let \(A\) and \(B\) be two sets then \((A \cup B)^{\prime} \cup\left(A^{\prime} \cap B\right)\) is equal to (A) \(B^{\prime}\) (B) \(B\) (C) \(A\) (D) \(A^{\prime}\)

If \(A\) is the set of even natural numbers less than 8 and \(\mathrm{B}\) is the set of prime numbers less then 7 , then the number of relations from \(A\) to \(B\) is (A) \(2^{9}\) (b) \(9^{2}\) (C) \(3^{2}\) (D) \(2^{9}-1\)

If \(P, Q\) and \(R\) are subsets of a set \(\mathrm{A}\), then \(R \times\left(P^{\prime} \cup Q^{\prime}\right)^{\prime}\) equals (A) \((R \times P) \cap(R \times Q)\) (B) \((R \times Q) \cap(R \times P)\) (C) \((R \times P) \cup(R \times Q)\) (D) None of these

For real numbers \(x\) and \(y\), define a relation \(R, x R y\) if and only if \(x-y+\sqrt{2}\) is an irrational number. Then the relation \(R\) is (A) reflexive (B) symmetric (C) transitive (D) an equivalence relagtion

If \(\mathrm{A}=\left\\{x: x^{2}=1\right\\}\) and \(B\left\\{x: x^{4}=1\right\\}\), then \(A \Delta B\) is equal to: (A) \(\\{i,-i\\}\) (B) \(\\{-1,1\\}\) (C) \(\\{-1,1, i,-i\\}\) (D) None of these

Which of the following is a singleton set? (A) \(\\{x:|x|<1, x \in Z\\}\) (B) \(\\{x:|x|=5, x \in \mathrm{Z}\\}\) (C) \(\left\\{x: x^{2}=1, x \in Z\right\\}\) (D) \(\left\\{x: x^{2}+x+1=0, x \in R\right\\}\)

Consider the following relations: (1) \(A-B=A-(A \cap B)\) (2) \(A=(A \cap B) \cup(A-B)\) (3) \(A-(B \cup C)=(A-B) \cup(A-C)\) Which of these is/are correct? (A) 1 and 3 (B) 2 only (C) 2 and 3 (D) 1 and 2

Let \(R\) be a reflexive relation on a finite set \(A\) having \(n\) elements, and let there be \(m\) ordered pairs in \(R\). Then (A) \(m \geq n\) (B) \(m \leq n\) (C) \(m=n\) (D) None of these

If two sets \(A\) and \(B\) are having 99 elements in common then the number of elements common to each of the sets \(A \times B\) and \(B \times A\) are (A) \(2^{99}\) (B) \(99^{2}\) (C) 100 (D) 18 (E) 9

The relation \(R\) defined on the set \(A=[1,2,3,4,5]\) by \(R=\left[(x, y):\left|x^{2}-y^{2}\right|<16\right]\) is given by, (A) \([(1,1),(2,1)(3,1),(4,1),(2,3)]\) (B) \([(2,2),(3,2)(4,2),(2,4)]\) (C) \([(3,3),(3,4)(5,4),(4,3),(3,1)]\) (D) None of these

Let \(L\) denotes the set of all straight lines in a plane. Let a relation \(R\) be defined by \(\alpha R \beta \Leftrightarrow \alpha \perp \beta, \alpha, \beta \in L\) Then \(R\) is (A) reflexive (B) symmetric (C) transitive (D) None of these

Let \(R=[(2,3),(3,4)]\) be a relation defined on the set of natural numbers. The minimum number of ordered pairs required to be added in \(R\) so that enlarged relation be comes an equivalence relation is (A) 3 (B) 5 (C) 7 (D) 9

Let \(R\) be the set of real numbers. Statement 1: \(A=\\{(x, y) \in R \times R: y-x\) is an integer \(\\}\) is an equivalence relation of \(R\). Statement \(2: B=\\{(x, y) \in R \times R: x=\alpha y\) for some rational number \(\alpha\\}\) is an equivalence relation of \(R\). (A) Statement 1 is false, Statement 2 is true (B) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1 (C) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (D) Statement 1 is true, Statement 2 is false

Let \(X=\\{1,2,3,4,5\\} .\) The number of different ordered pairs \((Y, Z)\) that can formed such that \(Y \subseteq X, Z \subseteq X\) and \(Y \cap Z\) is empty, is (A) \(5^{2}\) (B) \(3^{5}\) (C) \(2^{5}\) (D) \(5^{3}\)

Let \(R\) be the real line. Consider the following subsets of the plane \(R \times R\). \(S=[(x, y): y=x+1\) and \(0

Let \(R=\\{(1,3),(4,2),(2,4),(2,3),(3,1)\\}\) be a relation on the set \(A=\\{1,2,3,4\\} .\) The relation \(R\) is \(\quad\) [2004] (A) a function (B) reflexive (C) not symmetric (D) transitive

Let \(R=\\{(3,3),(6,6),(9,9),(12,12),(6,12),(3,9)\) \((3,12),(3,6)\\}\) be a relation on the set \(A=\\{3,6,9,12\\}\) be a relation the set \(A=\\{3,6,9,12\\}\). The relation is [2005] (A) reflexive and transitive only (B) reflexive only (C) an equivalence relation (D) reflexive and symmetric only

Let \(W\) denote the words in the English dictionary. Define the relation \(R\) by: \([2006]\) \(R=\\{(x, y) \in W \times W \mid\) the words \(x\) and \(y\) have at least one letter in common\\}. Then \(R\) is (A) not reflexive, symmetric and transitive (B) reflexive, symmetric and not transitive (C) reflexive, symmetric and transitive (D) reflexive, not symmetric and transitive

The set \(S=\\{1,2,3, \ldots, 12\) ) is to be partitioned into three sets \(A, B, C\) of equal size. Thus, \(A \cup B \cup C=S\), \(A \cap B=B \cap C=A \cap C=\phi .\) The number of ways to partition \(\mathrm{S}\) is (A) \(\frac{12 !}{3 !(4 !)^{3}}\) (B) \(\frac{12 !}{3 !(3 !)^{4}}\) (C) \(\frac{12 !}{(4 !)^{3}}\) (D) \(\frac{12 !}{(3 !)^{4}}\)

Let \(R\) be the real line. Consider the following subsets of the plane \(R \times R\). \(S=\\{(x, y): y=x+1\) and \(0

If \(A, B\) and \(C\) are three sets such that \(A \cap B=A \cap C\) and \(A \cup B=A \cup C\), then (A) \(A=B\) (B) \(A=C\) (C) \(B=C\) (D) \(A \cap B=\phi\)

Let \(S\) be a non-empty subset of \(R\). Consider the following statement: \(P\) : There is a rational number \(x \in S\) such that \(x>0\). Which of the following statements is the negation of the statement \(P\) ? (A) There is no rational number \(x \in S\) such that \(x \leq 0\) (B) Every rational number \(x \in S\) satisfies \(x \leq 0\) (C) \(x \in S\) and \(x \leq 0 \Rightarrow x\) is not rational (D) There is a rational number \(x \in S\) such that \(x \leq 0\)

Let \(R\) be the set of all real numbers. Statement 1: \(A=\\{(x, y) \in R \times R: y-x\) is an integer \(\\}\) is an equivalence relation on \(R\). Statement 2: \(B=\\{(x, y) \in R \times R: x=\alpha y\) for some rational number \(\alpha\\}\) is an equivalence relation on \(R\). (A) Statement 1 is true, Statement 2 is true; Statement 2 is not a correct explanation for Statement 1 (B) Statement 1 is true, Statement 2 is false. (C) Statement 1 is false, Statement 2 is true. (D) Statement 1 is true, Statement 2 is true; Statement 2 is a correct explanation for Statement 1

Let \(A\) and \(B\) be two sets containing 2 elements and 4 elements respectively. The number of subsets of \(A \times B\) having 3 or more elements is (A) 220 (B) 219 (C) 211 (D) 256

Let \(A\) and \(B\) be two sets containing four and two elements respectively. Then the number of subsets of the set \(A \times B\), each having at least three elements is: \([2015]\) (A) 256 (B) 275 (C) 510 (D) 219