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Algebra

Sets, Relations and Function: JEE Main & Advanced PYQs

214 previous year questions on Sets, Relations and Function from JEE Main and JEE Advanced papers, each with its exam and year, an answer and a solution.

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Questions 151-200 of 214

  1. Let $f(x)=x^{2}$, $x\in\mathbb R$. For any $A\subseteq\mathbb R$, define $g(A)=\{x\in\mathbb R: f(x)\in A\}$. If $S=[0,4]$, then which one of the following statements is …
  2. Define a relation R on the interval $\left[0,\frac\pi2\right)$ by $x\mathrm{R}y$ if and only if $\sec^2x-\tan^2y=1$. Then R is :
  3. Let $A$, $B$ and $C$ be sets such that $\phi\neq A\cap B\subseteq C$. Then which of the following statements is not true?
  4. Among the relations $S=\left\{(a,b): a,b\in R-\{0\},\ 2+\frac ab>0\right\}$ and $T=\left\{(a,b): a,b\in R,\ a^{2}-b^{2}\in Z\right\}$,
  5. The negation of the Boolean expression ${\sim}s\vee\left({\sim}r\wedge s\right)$ is equivalent to
  6. Which of the following statements is a tautology?
  7. The Boolean expression $\sim(p\Rightarrow(\sim q))$ is equivalent to
  8. The compound statement $(\sim(P\wedge Q))\vee((\sim P)\wedge Q)\Rightarrow((\sim P)\wedge(\sim Q))$ is equivalent to
  9. Let $P(S)$ denote the power set of $S=\{1,2,3,\ldots,10\}$. Define the relations $R_1$ and $R_2$ on $P(S)$ as $AR_1B$ if …
  10. Let $Z$ be the set of integers. If $A=\left\{x\in Z: 2^{(x+2)(x^{2}-5x+6)}=1\right\}$ and $B=\{x\in Z: -3<2x-1<9\}$, then the number of subsets of the set $A\times B$ is …
  11. Consider the following two statements: $P$ : If $7$ is an odd number, then $7$ is divisible by $2$. $Q$ : If $7$ is a prime number, then $7$ is an odd number. If $V_1$ …
  12. If $R$ is the smallest equivalence relation on the set $\{1,2,3,4\}$ such that $\{(1,2),(1,3)\}\subset R$, then the number of elements in $R$ is :
  13. If $p\to(\sim p\vee\sim q)$ is false, then the truth values of $p$ and $q$ are, respectively
  14. Consider the following two binary relations on the set $A=\{a,b,c\}$ : $R_1=\{(c,a),(b,b),(a,c),(c,c),(b,c),(a,a)\}$ and …
  15. Let the set of all relations $R$ on the set $\{a,b,c,d,e,f\}$, such that $R$ is reflexive and symmetric, and $R$ contains exactly 10 elements, be denoted by …
  16. Let $N$ denote the set of all natural numbers. Define two binary relations on $N$ as $R_1=\{(x,y)\in N\times N: 2x+y=10\}$ and $R_2=\{(x,y)\in N\times N: x+2y=10\}$. Then
  17. If the truth value of the statement $p\to(\sim q\vee r)$ is false ($F$), then the truth values of the statements $p,q,r$ are respectively
  18. The negation of the expression $q\vee\left((\sim q)\wedge p\right)$ is equivalent to
  19. The minimum number of elements that must be added to the relation $R=\{(a,b),(b,c)\}$ on the set $\{a,b,c\}$ so that it becomes symmetric and transitive is:
  20. Let $R$ be a relation on $\mathbb R$, given by $R=\{(a,b):3a-3b+\sqrt7 \text{ is an irrational number}\}$. Then $R$ is
  21. The expression ${\sim}({\sim}p\to q)$ is logically equivalent to
  22. Let $X=\{n\in N:1\le n\le50\}$. If $A=\{n\in X:n\text{ is a multiple of }2\}$ and $B=\{n\in X:n\text{ is a multiple of }7\}$, then the number of elements in the smallest …
  23. Which one of the following Boolean expressions is a tautology?
  24. Let $a=3\sqrt{2}$ and $b=\frac{1}{5^{1/6}\sqrt{6}}$. If $x,y\in\mathbb{R}$ are such that $$3x+2y=\log_a\left(18\right)^{\frac{5}{4}}$$ and …
  25. Consider the following relations $R=\{(x,y)\mid x,y\text{ are real numbers and }x=wy\text{ for some rational number }w\}$ …
  26. Let $P$ be the relation defined on the set of all real numbers such that $P=\left\{(a,b):\sec^{2}a-\tan^{2}b=1\right\}$. Then, $P$ is
  27. Let $R$ be the real line. Consider the following subsets of the plane $R\times R$ : $S=\{(x,y):y=x+1\text{ and }0<x<2\}$ $T=\{(x,y):x-y\text{ is an integer}\}$ Which one …
  28. Let $R$ be the set of real numbers. Statement I: $A=\{(x,y)\in R\times R:y-x\text{ is an integer}\}$ is an equivalence relation on $R$. Statement II …
  29. Let $W$ denote the words in the English dictionary. Define the relation $R$ by …
  30. The set of all $x$ for which $\ln(1+x)\le x$ is equal to ______.
  31. Let $A=\{2,3,5,7,9\}$. Let $R$ be the relation on $A$ defined by $xRy$ if and only if $2x\le3y$. Let $l$ be the number of elements in $R$, and $m$ be the minimum number …
  32. The number of elements in the relation $R=\{(x,y):4x^2+y^2<52,\ x,y\in\mathbf{Z}\}$ is :
  33. Let $R$ be a relation defined on the set $\{1,2,3,4\}\times\{1,2,3,4\}$ by $R=\{((a,b),(c,d)):2a+3b=3c+4d\}$. Then the number of elements in $R$ is :
  34. Let $f(x)=[x]^2-[x+3]-3$, $x\in\mathbf{R}$, where $[\cdot]$ is the greatest integer function. Then
  35. Let $A=\{x:|x^2-10|\le6\}$ and $B=\{x:|x-2|>1\}$. Then
  36. Let the relation $R$ on the set $M=\{1,2,3,\ldots,16\}$ be given by $R=\{(x,y):4y=5x-3,\ x,y\in M\}$. Then the minimum number of elements required to be added in $R$, in …
  37. The number of relations, defined on the set $\{a,b,c,d\}$, which are both reflexive and symmetric, is equal to :
  38. Let $A=\{-2,-1,0,1,2,3,4\}$. Let $R$ be a relation on $A$ defined by $xRy$ if and only if $2x+y\le2$. Let $l$ be the number of elements in $R$. Let $m$ and $n$ be the …
  39. Let $A=\{0,1,2,\ldots,9\}$. Let $R$ be a relation on $A$ defined by $(x,y)\in R$ if and only if $|x-y|$ is a multiple of $3$. Given below are two statements. Statement …
  40. Consider two sets $A=\{x\in\mathbf{Z}:|(|x-3|-3)|\le1\}$ and $B=\left\{x\in\mathbf{R}-\{1,2\}:\frac{(x-2)(x-4)}{x-1}\log_e(|x-2|)=0\right\}$. Then the number of onto …
  41. Let $(x_0, y_0)$ be the solution of the following equations $$(2x)^{\ln 2}=(3y)^{\ln 3}$$ $$3^{\ln x}=2^{\ln y}.$$ Then $x_0$ is
  42. Let $m$ be the minimum possible value of $\log_3(3^{y_1}+3^{y_2}+3^{y_3})$, where $y_1,y_2,y_3$ are real numbers for which $y_1+y_2+y_3=9$. Let $M$ be the maximum …
  43. The value of $6+\log_{\frac32}\left(\dfrac{1}{3\sqrt2}\sqrt{4-\dfrac{1}{3\sqrt2}\sqrt{4-\dfrac{1}{3\sqrt2}\sqrt{4-\dfrac{1}{3\sqrt2}\cdots}}}\right)$ is
  44. The value of $$\left((\log_2 9)^2\right)^{\frac{1}{\log_2(\log_2 9)}}\times(\sqrt{7})^{\frac{1}{\log_4 7}}$$ is ______ .
  45. Let $X=\mathbf{R}\times\mathbf{R}$. Define a relation $R$ on $X$ as : $(a_1,b_1)\ R\ (a_2,b_2)\Leftrightarrow b_1=b_2$. Statement I : $R$ is an equivalence relation. …
  46. Let $A=\{(x,y)\in\mathbf{R}\times\mathbf{R}:|x+y|\ge 3\}$ and $B=\{(x,y)\in\mathbf{R}\times\mathbf{R}:|x|+|y|\le 3\}$. If $C=\{(x,y)\in A\cap B: x=0\text{ or }y=0\}$ …
  47. The relation $R=\{(x,y): x,y\in\mathbb{Z}\text{ and }x+y\text{ is even}\}$ is :
  48. 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 :
  49. Let $A$ and $B$ be sets containing four and two elements respectively. Then, the number of subsets of the set $A\times B$, each having at least three elements are
  50. The least value of the expression $2\log_{10}x-\log_{x}(0.01)$, for $x>1$, is

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