Set theory practice · Mixed difficulty

Let R1, R2 are relations defined on Z such that a $a{R_1}b \Leftrightarrow (a - b)$ is divisible by 3 and a R2 $b \Leftrightarrow (a - b)$ is divisible by 4. Then which of the two relations $\left( {{R_1} \cup {R_2}} \right),\left( {{R_1} \cap {R_2}} \right)$ is an
equivalence relation?

If $A$ and $B$ be two sets containing 3 and 6 elements respectively, what can be the minimum number of elements in $A \cup B$ ?

If $A=\{y: y=2 x, x \in N\}, B=\{y: y=2 x-1, x \in N$, then $(A \cap B)^{\prime}$ is

Which one of the following is correct ?

If $A=\{x: x=2 n+1, n \in Z$ and $B=\{x: x=2 n, n \in Z\}$, then $A \cup B$ is

For any three sets $A, B$ and $C$ the set $(A \cup B \cup C) \cap\left(A \cap B^{\prime} \cap C^{\prime}\right)^{\prime} \cap C^{\prime}$ is equal to

If $A = \left\{ {1,\;2,\;3,\;4} \right\},$ then the number of subsets of set A containing element 3, is

If $X$ and $Y$ are two sets, then $X \cap {\left( {Y \cup X} \right)^\prime }$ equals

If $A=\{1,2\}, B=\{1,3\}$, then $(A \times B) \cup(B \times A)$ is equal to

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 survey shows that 63% of the Americans like cheese whereas 76% like apples. If $x\%$ of the Americans like both cheese and apples, then

The number of non-empty subsets of the set $\{1,2,3,4\}$ is

If $P$ is the set of all parallelograms, and $T$ is the set of all trapeziums, then $P \cap T$ is

Which of the following is/are true?

I. If $A$ is a subset of the universal set $U$, then its complement $\mathrm{A}^{\prime}$ is also a subset of $\mathrm{U}$.

II. If $\mathrm{U}=\{1,2,3, \ldots ., 10\}$ and $\mathrm{A}=\{1,3,5$, $7,9\}$, then $\left(A^{\prime}\right)^{\prime}=A$.

If $A=\left\{x: x^2-5 x+6=0\right\}, B=\{2,4\}, C=\{4,5\},$ then $A \times(B \cap C)$ is

The intersection of all the intervals having the form $\left[1+\frac{1}{n}, 6-\frac{2}{n}\right]$, where $n$ is a positive integer is

Let $\mathrm{S}=$ the set of all triangles, $\mathrm{P}=$ the set of all isosceles triangles, $Q=$ the set of all equilateral triangles, $\mathrm{R}=$ the set of all right-angled triangles.
What do the sets $P \cap Q$ and $R-P$ represents respectively ?

Let $S=\left\{x \in[-6,3]-\{-2,2\}: \frac{|x+3|-1}{|x|-2} \geq 0\right\}$ and $T=\left\{x \in \mathbb{Z}: x^2-7|x|+9 \leq 0\right\}$. Then the number of elements in $S \cap T$ is

Let $Z$ denote the set of all integers and $A = \left\{ {\left( {a,b} \right):{a^2} + 3{b^2} = 28,\;\;a,\;b \in Z} \right\}$ and $B = \{ \left( {a,\;b} \right):a > b,\;a,\;b \in Z\}$. Then, the number of elements in $A \cap B$ is

In a town of 10,000 families it was found that $40 \%$ family buy newspaper A, $20 \%$ buy newspaper $\mathrm{B}$ and $10 \%$ families buy newspaper C, $5 \%$ families buy A and B, $3 \%$ buy $\mathrm{B}$ and $\mathrm{C}$ and $4 \%$ buy $\mathrm{A}$ and $\mathrm{C}$. If $2 \%$ families buy all the three newspapers, then number of families which buy A only is

If $N_a=\{a n: n \in N\}$, then $N_5 \cap N_7=$

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 $\cap _{i = 3}^{100}{A_i} = A,$ then $n\left( A \right) =$

If $A=\left\{x \in \mathbb{C}: x^2=1\right\}$ and $B=\left\{x \in \mathbb{C}: x^4=1\right\}$, then $A \Delta B$ is equal to

If $A \subseteq B$, then $B \cup A$ is equal to

Which of the following is a singleton set?

If the sets $A$ and $B$ are defined as:
$\begin{aligned} & A=\left\{(x, y): y=e^x, x \in R\right\} \\ & B=\{(x, y): y=x, x \in R\}, \text { then } \end{aligned}$

In a school, there are 20 teachers who teach Mathematics or Physics of these, 12 teach Mathematics and 4 teach both Maths and Physics. Then the number of teachers teaching only Physics are

In a town of 10,000 families it was found that 40% families buy newspaper A,20% families buy newspaper B and 10% families buy newspaper C,5% families buy A and B, 3% buy B and C and 4% buy A and C. If 2% families buy all the three newspapers, then the number of families which buy A only is

Let ${A}=\{2,3,4\}$ and ${X}=\{0,1,2,3,4\}$, then which of the following statements is correct

If $U=\{1,2,3,4,5,6,7,8,9\}, A=\{2,4,6,8\}$ and $B=\left\{2,3,5,7\right.$, then $(A \cup B)^{\prime},\left(A^{\prime} \cap B^{\prime}\right)(A \Delta B)$ is equal to