Mathematics

Set Theory and Relations

355 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 maths fun with numbers some special sequences triangular numbers properties and patterns of perfect squares

Let $S$ be the set of all ordered pairs $(x,y) $ of positive integers satisfying the condition $x^{2}-y^{2}=12345678$. Then:

  1. $S$ is an infinite set
  2. $S$ is the empty set
  3. $S$ has exactly one element
  4. $S$ is a finite set and has at least two elements
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$x^{2}-y^{2}=12345678 (x,y \, \epsilon \,  1^{+})$

RHS is even, so, x & y should be odd integer but difference square of two odd integer is multiple of 8 but RHS is not multiple of $8$

$\therefore  8 $ is an empty set.
Multiple choice mathematics and statistics relations cartesian product of sets cartesian product of two sets cartesian product

If $A$ and $B$ are 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

  1. $219$
  2. $256$
  3. $275$
  4. $510$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$A=\left{a,b,c,d\right}$

$B=\left{x,y\right}$
$A\times B=2\times 4=8$ Elements.
Total number of subsets of $A\times B$ having $3$ or more elements $=2^8=256$
$\Rightarrow 256-(1$ null set $+$ $8$ singleton set $+$ $^8e _2$having $2$ elements$)$
$=256-1-8-28$
$=219$

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.