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 introduction to set union and intersections union and intersection of sets basic operations on sets

Let $A=\left{ a,b,c,d \right} ,B=\left{ b,c,d,e \right}$. Then $n\left[ \left( A\times B \right) \cap \left( B\times A \right)  \right]$ is equal to 

  1. $3$
  2. $6$
  3. $9$
  4. $none$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The number of elements in the Cartesian product of two sets is the product of their cardinalities. Since both sets A and B have 4 elements, the intersection of A cross B and B cross A consists of the ordered pairs (x, y) where both x and y belong to the intersection of A and B. The intersection of A and B contains 3 elements, so the number of such pairs is 3 squared, which is 9.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let A={1, 2, 3, 4), B={2, 3, 4, 5}, then $n{ (A\times B)\cap (B\times A)} =$?

  1. 13

  2. 16

  3. 9

  4. 10
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The sets A and B share three common elements, namely 2, 3, and 4. The intersection of A cross B and B cross A is equivalent to the Cartesian product of the intersection of A and B with itself. Since the intersection has 3 elements, its square has 3 times 3, or 9 elements.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A = {1, 2, 3, 4, 5}, B = {2, 4, 6, 8}$ and C= ${3,4,5,6}$, 

then verify : $A - (B \cup C) = (A - B) \cap (A - C)$.

  1. True

  2. False
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given, $A = \{1, 2, 3, 4, 5\}, B = \{2, 4, 6, 8\}$ and $C=\{3,4,5,6\}$

For the LHS:

Union of two sets will have the elements of both sets.

So, $ B \cup C = \{2,3,4,5,6,8 \}$ 

$ A - (B \cup C) $ will have elements of $A$ which are not in $ (B \cup C) $

So, $ A - (B \cup C) = \{ 1 \}$ ..... $(1)$

For the RHS:

$ A - B $ will have elements of $A$ which are not in $B$.

So, $ A - B = \{ 1,3,5 \}$  

$ A - C $ will have elements of $A$ which are not in $C$.

So, $ A - C = \{ 1,2 \}$  

Intersection of two sets has the common elements of both the sets. 

$\Rightarrow (A - B) \cap (A - C) = \{1\}$ ..... $(2)$

From $(1)$ and $(2),$ we have

$ A - (B \cup C) =(A - B) \cap (A - C) $

Hence, the given expression is true.
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $P _1$ be the set of all prime numbers, i.e., $P _1=\left {2, 3, 5, 7, 11, ....\right }$, Let $Pn=\left {np|p\epsilon P _1|\right }$, i.e., the set of all prime multiples of n. Then which of the following sets is non empty?

  1. $P _1\cap P _{23}$
  2. $P _7\cap P _{21}$
  3. $P _{12}\cap P _{20}$
  4. $P _{20}\cap P _{24}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Check by option
$P _{12}=\left {24, 36, 60, 84, ....\right }$
$P _{20}=\left {40, 60, 100, .....\right }$
$P _{12}\cap P _{20}$ has common element.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let A = {x : x is a square of a natural number and x is less than 100} and B is a set of even natural numbers. What is the cardinality of $ A \cap B$ ?

  1. 4

  2. 5

  3. 9

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

$A = {1, 4. 9. 16, 25, 36, 49, 64, 81 }$

$B = {2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 24, 26, 28, 30, 32, 34, 36, 38,...}$
$A \cap B = {4, 16, 36, 64}$
Hence, $n(a \cap B) = 4$.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Two set A and B are as under 
A = {(a,b) $\epsilon$ R $\times$ R : $\mid a - 5\mid$ < $1$ and  $\mid b - 5\mid$ < $1$};
B = {(a,b) $\epsilon$ R $\times$ R : $4(a-6)^2 + 9(b-5)^2$ $\leq 36$. Then, 

  1. B $\subset$ A
  2. A $\subset$ B
  3. A $\bigcap$ B = $\phi$ (an empty set)
  4. nither A $\subset$ B nor B $\subset$ A
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Set A is a square centered at (5, 5) with side length 2. Set B is an ellipse centered at (6, 5) with semi-axes 3 and 2. By comparing the boundaries, one can show that all points in A satisfy the inequality for B.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

$R$ is the set of all positive odd integers less than $20$; $S$ is the set of all multiples of $3$ that are less than $20$. How many elements are in the set $R$ $\cap$ $S$?

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $R=$ {$1,3,5,7,9,11,13,15,17,19$} , $S=$ {$3,6,9,12,15,18$}
Therefore the intersection of $S$ and $R$ is {$3,9,15$}.
So the number of elements which are common to both is $3$.

Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Let $Z$ denotes 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{ \left( a,b \right) :a < b,a,b\in Z \right} $. Then, the number of elements in $A\cap B$ is

  1. $2$
  2. $4$
  3. $6$
  4. $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\because A=\left{ \left( a,b \right) :{ a }^{ 2 }+3{ b }^{ 2 }=28,a,b\in Z \right} $
   $=\left{ \left( 5,1 \right) ,\left( -5,-1 \right) ,\left( 5,-1 \right) ,\left( -5,1 \right) ,\left( 4,2 \right) ,\left( -4,-2 \right) ,\left( 4,-2 \right) ,\left( -4,2 \right) ,\left( 1,3 \right) ,\left( -1,-3 \right) ,\left( 1,-3 \right) ,\left( -1,3 \right)  \right}$
and $B=\left{ \left( a,b \right) :a<b,a,b\in Z \right} $
$\therefore A\cap B=\left{ \left( 1,3 \right) ,\left( -1,3 \right) ,\left( -4,-2 \right) ,\left( -4,2 \right) ,\left( -5,-1 \right) ,\left( -5,1 \right)  \right} $
$\therefore $ The number of elements in $A\cap B$ is $6$.

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$)$
$=256-1-8-28$
$=219$