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

If $aN=\left{ ax:x\epsilon N \right}$, then the set $3N\cap 7N$ is

  1. $21\ N$
  2. $10\ N$
  3. $4\ N$

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

The set nN represents multiples of n. The intersection of multiples of 3 (3N) and multiples of 7 (7N) consists of multiples of the least common multiple of 3 and 7, which is 21. Thus, 3N intersect 7N = 21N.

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

A is a set containing $n$ elements. $A$ subset $P$ of $A$ is chosen. the set $A$ is reconstructed by replacing the elements of $P.A$ subset $Q$ of $A$ is again chosen. the number of ways of choosing $P$ and $Q$ so that $P \cap Q$

  1. $9. ^{n}C _{2}$
  2. $3^{n}- ^{n}C _{2}$
  3. $^{n}C _{2}.3^{n-2}$
  4. $4^{n}-3^{n}$
Reveal answer Fill a bubble to check yourself
A Correct answer
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 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 $6$.