Set theory practice · Mixed difficulty

The number of elements in the set $\left\{ {\left( {a,b} \right):2{a^2} + 3{b^2} = 35,\;a,\;b \in Z} \right\}$, where $Z$ is the set of all integers, is

For any two sets $A$ and $B$, if $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$ for some set $X$, then

Three sets $A, B, C$ are such that $A=B \cap C$ and $B=C \cap A$, then

I. The collection of all months of a year beginning with the letter J.

II. The collection of ten most talented writers of India.

III. A team of eleven best cricket batsmen of the world.

IV. The collection of all boys in your class.

Which of the above are the sets?

In a committee 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. The number of persons speaking at least one of these two languages is

Consider the following statements.

I. If (a, 1), (b, 2) and (c, 1) are in $\mathrm{A} \times \mathrm{B}$ and $\mathrm{n}(\mathrm{A})$ $=3, n(B)=2$, then $A=\{a, b, c\}$ and $B=\{1,2\}$

II. If $A=\{1,2\}$ and $B=\{3,4\}$, then $A \times(B \cap \phi)$ is equal to $\mathrm{A} \times \mathrm{B}$.

Choose the correct option.

Which of the following is the empty set?

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

Let $\mathrm{A}=\{\mathrm{x}: \mathrm{x} \in \mathrm{R},-1<\mathrm{x}<1\}, \mathrm{B}=\{\mathrm{x}: \mathrm{x} \in \mathrm{R}, \mathrm{x} \leq 0$ or $\mathrm{x} \geq 2\}$ and $\mathrm{A} \cup \mathrm{B}=\mathrm{R}-\mathrm{D}$, then the set $\mathrm{D}$ is

The empty set is represented by

I. $\phi \quad$ II. $\{\phi\} \quad$ III. $\{\} \quad$ IV. $\{\{\}\}$

If $a\;N = \left\{ {a\;x\;:x \in N} \right\}$ and $b\;N \cap c\;N = d\;N,$ where $b,c \in N$ then

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

In the above question, the number of families which buy none of $A,B$ and $C$ is

$A=\left\{(x, y): x^2+y^2 \leq 9\right\}$

$B=\left\{(x, y):\left|x-1-x^2\right| \leq\left|x^2-3 x+4\right|\right\} \text { where }$

$x$ and $y$ are integers.

Number of integral $(\mathrm{x}, \mathrm{y})$ in set $\mathrm{A} \cap \mathrm{B}$ are

If $A$ and $B$ are two disjoint sets, then which one of the following is correct?

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 $\mathrm{D}$ is:

Suppose $n(A)=3, n(B)=6$ and $A \subseteq B$. Then, the number of elements in $A \cup B$ is equal to

The set $\{x: x$ is a positive integer less than 6 and $3^x-1$ is an even number $\}$ in roster form is

Which one of the following is correct?

The set of all letters of the word ‘SCHOOL’ is
represented by

I. {S, C, H, O, O, L}

II. {S, C, H, O, L}

III. {C, H, L, O, S}

IV. {S, C, H, L}

The correct code is

Let $\bigcup\limits_{i=1}^{50} X_i=\bigcup\limits_{j=1}^n Y_j=Z$, where each $X_i$ contains 10 elements and each $Y_j$ contains 5 elements. If each element of $Z$ is an element of exactly 20 of sets $X_i^{\text {'s }}$ and exactly 6 of sets $Y_j^{\text {'s }}$, then $n$ is equal to

If ${A_n}$ is the set of first $n$ prime numbers, then $\cup _{n = 2}^{10}{A_n} =$

Let $A, B, C$ be three sets such that $A \cup B \cup C=U$, where $U$ is the universal set. Then, ${\{ (A - B) \cup (B - C) \cup (C - A)\} ^\prime }$

Consider the following statements.

I. The vowels in the English alphabet.

II. The collection of books.

III. The rivers of India.

IV. The collection of most talented batsmen of India.

Which of the following is/are well-defined collections?

$A,B$ and $C$ are three non-empty sets. If $A \subset B$ and $B \subset C,$ then which of the following is true?

20 teachers of a school either teach Mathematics or Physics. 12 of them teach Mathematics, while 4 teach both the subject. Then, the number of teachers teaching Physics only is

Let $X$ be a family of sets and $R$ be a relation on $X$ defined by '$A$ is disjoint from $B'$. Then, $R$ is

If ${A}=\{1,3,5,7,9,11,13,15,17\}, {B}=\{2,4, \ldots 18\}$ and ${N}$ is the universal set then $A^{\prime} \cup\left(A \cup\left(B \cap B^{\prime}\right)\right)$ is

Two finite sets have m and n elements. The number of elements in the power set of first set is 48 more than the total number of elements in the power set of the second set. Then, the value of $M$ and $N$ are

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?