Mathematics

Set Theory and Relations

368 Questions

Set theory involves the study of collections of objects and includes operations like union, intersection, and finding complements. Questions cover power sets, Cartesian products, and properties of empty sets. This foundational mathematical topic frequently appears in various competitive exams and university entrance tests.

set operationspower setscartesian productsset complementsproperties of empty setsset partitions

Set Theory and Relations Questions

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A = {a, b}, B={1, 2, 3}$, find B $\times$ A

  1. $B$ $\times$ $A$$ = \{(1, a), (2, a), (3, a), (1, b) (2, b), (3, b)\}$
  2. $B$ $\times$ $A$$ = \{ (2, a), (3, a), (1, b) (2, b), (3, b)\}$
  3. $B$ $\times$ $A$$ = \{(1, a), (2, a), (3, a), (1, b) (2, b)\}$

  4. None of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

To find B × A multiply each element of B with that of A & form an ordered pair.

 i.e. ordered pairs are (1,a); (2,a); (3,a); (1,b); (2,b); (3,b)
Therefore B × A = {  (1,a), (2,a), (3,a), (1,b), (2,b), (3,b)}

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A = {2, 3, 5}$ and $B = {5, 7}$, find the set with highest number of elements:

  1. $A \times B$
  2. $ B \times A$
  3. $A \times A$
  4. $B \times B$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$A=\left \{ 2,3,5 \right \}$

$B=\left \{ 5,7 \right \}$

$A\times B=\left \{ (2,5),(2,7),(3,5),(3,7),(5,5),(5,7) \right \}$

$B\times A=\left \{ (5,2),(5,3),(5,5),(7,2),(7,3),(7,5) \right \}$

$A\times A=\left \{ (2,2),(2,3),(2,5),(3,2),(3,3),(3,5),(5,2),(5,3),(5,5) \right \}$

$B\times B=\left \{ (5,5),(5,7),(7,5),(7,7) \right \}$

$\therefore A\times A$ has the highest number of elements
Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

For two sets $A$ and $B$, $A\times B=B\times A$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

The given statement is false.

Example:-
Let us consider $A={1,2}$ and $B={3,4}$.

Now $A\times B={(1,3),(1,4),(2,3),(2,4)}$......(1).

And $B\times A={(3,1),(3,2),(4,1),(4,2)}$..........(2).

Form (1) and (2) it's evident that $A\times B\ne B\times A$.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let A and B be sets containing 2 and 4 elements respecetively. The number of subsets $A \times B$ having 3 or more elements is 

  1. $219$
  2. $211$
  3. $256$
  4. $220$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Let $A=\left\{x,y\right\}$
$B=\left\{a,bc,d\right\}$
$A\times B$ has $2\times 4=8$ elements
Total substance of $A\times B={2}^{8}=256$
$\therefore\,$Total number of subsets of $A\times B$ having $3$ or more elements
$=256-\left(1\,null \,set+8\,single\,ton\,set-^{8}C _{2}\,having\,2\,elements\right)$
$=256-1-8-\dfrac{8!}{6!2!}$
$=256-1-8-\dfrac{8\times 7\times 6!}{6!2!}$
$=256-1-8-28=219$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A={1, 2, 3}$ and $B={3, 8}$, then $(A\cup B)\times (A\cap B)$ is

  1. $\{(3, 1), (3, 2), (3, 3), (3, 8)\}$
  2. $\{(1, 3), (2, 3), (3, 3), (8, 3)\}$
  3. $\{(1, 2), (2, 2), (3, 3), (8, 8)\}$
  4. $\{(8, 3), (8, 2), (8, 1), (8, 8)\}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$A\cup B={1,2,3,8}$
$A\cap B={3}$
$\therefore (A\cup B)\times (A\cap B)$
$={(1,3),(2,3),(3,3),(8,3)}$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $A=\left { 1,2,3 \right }$ and $B=\left { a,b \right }$.Which of the following subsets of $A\times B$ is a mapping from $A$ to $B$

  1. $\left \{ \left ( 1,a \right ),\left ( 3,b \right ),\left ( 2,a \right ),\left ( 2,b \right ) \right \}$
  2. $\left \{ \left ( 1,b \right ),\left ( 2,a \right ),\left ( 3,a \right ) \right \}$
  3. $\left \{ \left ( 1,a \right ),\left ( 2,b \right ) \right \}$

  4. none of these
Reveal answer Fill a bubble to check yourself
A,B,C Correct answer
Explanation

$A=\left{ 1,2,3 \right} \ B=\left{ a,b \right} \ A\times B=\left{ \left( 1,a \right) ,\left( 2,a \right) ,\left( 3,a \right) ,\left( 1,b \right) ,\left( 2,b \right) ,\left( 3,b \right)  \right} .$


$ \left{ \left( 1,a \right) ,\left( 3,b \right) ,\left( 2,a, \right) \left( 2,b \right)  \right} \subset A\times B$

$ \left{ \left( 1,b \right) ,\left( 2,a \right) ,\left( 3,a \right)  \right} \subset A\times B$

$ \left{ \left( 1,a \right) ,\left( 2,b \right)  \right} \subset A\times B$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let $ A= { 1,2,3,.......50} $ and $B={2,4,6.......100}$ .The number of elements $\left ( x, y \right )\in A\times B$ such that $x+y=50$

  1. $24$
  2. $25$
  3. $50$
  4. $75$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The elements will be
$(2,48),(48,2)$
$(4,46), (46,4)$
:
:
$(2n,50-2n), (50-2n,2n)$
Now we have
$2,4,6,8...$ upto $48$
This forms an A.P
The number terms is $24$.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

Let A and B be two sets such that $A\times B=\left{ \left( a,1 \right) ,\left( b,3 \right) ,\left( a,3 \right) ,\left( b,1 \right) ,\left( a,2 \right) ,\left( b,2 \right)  \right} ,$ then 

  1. $A=\left\{ 1,2,3 \right\} $ and $B=\left\{ a,b \right\} $
  2. $A=\left\{ a,b \right\} $ and$ B=\left\{ 1,2,3 \right\} $
  3. $A=\left\{ 1,2,3 \right\} $ and $B\subset \left\{ a,b \right\} $
  4. $A\subset \left\{ a,b \right\} $ and $B\subset \left\{ 1,2,3 \right\} $
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$A$ is the first  element in  the cartesian product $A\times B=\left{a\,,b\,,a\,,b\,,a\,,b\right}$

and $B$ is the second element in  the cartesian product $A\times B=\left{1,\,3\,,3\,,1\,,2\,,2\right}$
$\therefore$ elements of $A=\left{a,b\right}$ and $B=\left{1\,,2\,,3\right}$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A = \left{2,3\right}$ and $B = \left{1,2\right}$, then $A \times B$ is equal to 

  1. $\left\{(2,1), (2,2), (3,1), (3,2)\right\}$
  2. $\left\{(1,2), (1,3), (2,2), (2,3)\right\}$
  3. $\left\{(2,1), (3,2)\right\}$
  4. $\left\{(1,2), (2,3)\right\}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If $A$ and $B$ are any two non-empty sets.
then $A\times B$$=\left{(x,y):x\in A  and  y\in B\right}$
As $A = \left{2,3\right}$ and $B = \left{1,2\right}$
$A \times B$$=\left{(2,1), (2,2), (3,1), (3,2)\right}$
Hence, option A.

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$, then $(A\cup B)\times (A\cap B)$ is equal to

  1. $\left \{(8,3), (8,2), (8,1), (8,8)\right \}$
  2. $\left \{(1,2), (2,2), (3,3), (8,8)\right \}$
  3. $\left \{(3,1), (3,2), (3,3), (3,8)\right \}$
  4. $\left \{(1,3), (2,3), (3,3), (8,3)\right \}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $A=\left {1, 2,3\right }$ and $B=\left {3,8\right }$,
Therefore, $A\cup B=\left {1,2,3\right }\cup \left {3,8\right }=\left {1,2,3,8\right }$
and $A\cap B=\left {1,2,3\right }\cap \left {3,8\right }=\left {3\right }$
$\therefore (A\cup B)\times (A\cap B)=\left {1,2,3,8\right }\times \left {3\right }$
$=\left {(1,3), (2,3), (3,3), (8,3)\right }$

Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

What is the Cartesian product of $A = \left {1, 2\right }$ and $B = \left {a, b\right }$?

  1. $\left \{(1, a), (1, b), (2, a), (b, b)\right \}$
  2. $\left \{(1, 1), (2, 2), (a, a), (b, b)\right \}$
  3. $\left \{(1, a), (2, a), (1, b), (2, b)\right \}$
  4. $\left \{(1, 1), (a, a), (2, a), (1, b)\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

If $A $ and $B$ are two non empty sets, then the Cartesian product $A \times B$ is set of all ordered pairs $(a,b)$ such that $a\in A$ and $b\in B$.


Given $A ={1,2}$ and $B = {a,b}$

Hence $A\times B = {(1,a),(1,b),(2,a),(2,b)}$