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 maths set concepts finite and infinite sets types of sets set language

A finite set $S$ is given by $S={x:x\in N: x\le15}.$ Find the cardinality of its power set.

  1. 32952

  2. 16384

  3. 32768

  4. 16476
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$S$ is given by $S={x:x\in N: x\le15}$
$S={1,2,3,4,5,6,7,8,9,10,11,12,13,14,15}$
$\therefore n(S)=15$
The power set is set of all the possible subsets of $S.$
Number of elements in power set $=$ Total number of subsets
So, cardinality of power set $S=2^{15}=32768$

Multiple choice maths set concepts finite and infinite sets types of sets set language

Identify the type of set
$N = {x : x \in  N, x < 7}$ 

  1. Finite Set

  2. Null Set

  3. Infinite Set

  4. Singleton Set
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

We have to identify the type of set.


Any set which is empty or contains a definite and countable number of elements is called a finite set. 

Any Set that does not contain any element is called the Null set

For uncountable or indefinite numbers of elements are referred to as  infinite sets.

Singleton Set are those sets that have only a single element.


Given $N=\{x|x\in N, x<7 \}$

               $=\{1,2,3,4,5,6\}$ which has finite number of elements i.e. countable number of elements. So it is a finite set.

Therefore $N$ is a finite set.
Multiple choice maths set concepts finite and infinite sets types of sets set language

Identify the type of set
$B={x: x \epsilon W,x=2n }$ 

  1. Finite Set

  2. Null Set

  3. Infinite Set

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

We have to identify the type of set.

Given $B={x|x\in W, x=2n }$ where $W$ represents a set of whole numbers.

               $={0,2,4,6,8,....}$ which has infinite number of elements. So it is an infinite set.

Therefore $B$ is an infinite set.

Multiple choice maths set concepts finite and infinite sets types of sets set language

Identify the type of set
$A= { x| x \epsilon R, 2 \leq x \leq 3 }$ 

  1. Finite Set

  2. Infinite Set

  3. Null Set

  4. Singleton Set
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We have to identify the type of set.

Given $A={x|x\in R, 2 \leq x \leq 3 }$

$A$ contains all real numbers between $2$ and $3$ which is an infinite set.

Therefore $A$ is an infinite set

Multiple choice maths set concepts finite and infinite sets types of sets set language

State which of the following are infinite sets.
$(i)A={x:x\in Z: x^2 $ is even $}$
$(ii)B={x:x\in R:-4<x<-2}$

  1. $(i)$ only
  2. $(ii)$ only
  3. $(i)$ and $(ii)$ both
  4. Neither $(i)$ nor $(ii)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(i)A={x:x\in Z: x^2 $ is even $}$
$A={...,-6,-4,-2,0,2,4,6,...}$ which is an infinite set.
$(ii)B={x:x\in R:-4<x<-2}$
There will be infinite real numbers any two numbers so, it is an infinite set.

Multiple choice maths set concepts finite and infinite sets types of sets set language

Which of the following are infinite set?
$(i)$The set of lines which are parallel to x-axis.
$(ii)$The set of animals living on the earth.
$(iii)$ The set of numbers which are multiple of $5.$
$(iv)$ The set of the circles passing through the origin $(0,0).$

  1. $(i),(ii)$ and $(iv)$
  2. $(ii)$ only
  3. $(i),(iii)$ and $(iv)$
  4. $(i),(ii)$ and $(iii)$$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(i)$The set of lines which are parallel to x-axis is an infinite set because line parallel to x-axis are infinite in number.
$(ii)$The set of animals living on the earth is a finite set because the number of animals living on the earth is finite (although it is quite a big number)
$(iii)$ The set of numbers which are multiple of $5$ is an infinite numbers multiples of $5$ are infinite in number.
$(iv)$ The set of the circles passing through the origin $(0,0)$ is an infinite set because infinite number of circles can pass through the origin.

Multiple choice maths set concepts finite and infinite sets types of sets set language

Which of the following sets are finite sets.
$(i)$ The sets of months in a year.
$(ii){1,2,3,....}$
$(iii){1,2,3,...,99,100}$
$(iv)$ The set of positive integers greater than $100.$ 

  1. $(i)$ and $(iii)$
  2. $(i)$ only
  3. $(ii),(iii)$ and $(iv)$
  4. $(ii)$ and $(iv)$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$(i)$ The sets of months in a year is a finite set because it has $12$ elements.
$(ii){1,2,3,....}$ is an infinite set as it has infinite number of elements.
$(iii){1,2,3,...,99,100}$ is a finite set as it has number from $1$ to $100$ which is finite in number.
$(iv)$ The set of positive integers greater than $100$ is an infinite set because positive integers greater than $100$ are infinite in number. 

Multiple choice maths set concepts finite and infinite sets types of sets set language

State which of the following are infinite sets.
$(i)A={x:x\in Z: x $ is odd$}$
$(ii)B={x:x\in R:<-10}$

  1. $(i)$ only
  2. $(ii)$ only
  3. $(i)$ and $(ii)$ both
  4. Neither $(i)$ nor $(ii)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$(i)A={x:x\in Z: x^2 $ is even $}$
$A={...,-3,-1-1,3,...}$ which is an infinite set.
$(ii)B={x:x\in R:-2<x<-4}$
$B={...,-14,-13,-12,-11}$ so it is an infinite set.

Multiple choice maths introduction to set different sets de morgan's law de morgan's law for set theory
Given, universal set = {$x \,\,\epsilon\,\, Z$ : $- 6 < x \leq 6$}, N = {$n$ : $n$ is a non-negative number} and P = {$x$ : $x$ is a nonpositive number}. Find :$P'$
  1. $\{-1,-2,-3,-4,-5,-6\}$
  2. $\{1,2,3,4,5,6\}$
  3. $\{0,1,2,3,4,5,6\}$
  4. $\{0,-1,-2,-3,-4,-5,-6\}$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Universal set includes ${-5,-4,-3,-2,-1,0,1,2,3,4,5,6}$


P=${-5.-4.-3,-2,-1,0}$.

Hence P' only has the elements in option B. It does not contain 0.
P'=${1,2,3,4,5,6}$.

Multiple choice maths introduction to set different sets de morgan's law de morgan's law for set theory

If the universal set ${x\in W ,3<x≤12} ,A={5,7,9}$, then $A'=$

  1. $\{3,6,8,10,11,12\}$
  2. $\{4,6,8,10,11,12\}$
  3. $\{6,8,10,11,12\}$

  4. None of the above
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
Given universal set is $\{x\in W ,3<x≤12\}$
It can be written as a set $\{4,5,6,7,8,9,10,11,12\}$
$A$ is given as $A=\left\{5,7,9\right\}$
$\therefore A'=W-A=$ All the elements in the universal set but not in set $A$
         $ =\{4,6,8,10,11,12\}$
Hence, option B is correct
Multiple choice maths introduction to set different sets de morgan's law de morgan's law for set theory

If the universal set is U = $ \displaystyle \left { 1^{2},2^{2},3^{2},4^{2},5^{2},6^{2} \right }  $   What is the complement of the intersection of set A = $ \displaystyle \left { 2^{2},4^{2},6^{2} \right }  $ and set B=$ \displaystyle \left { 2^{2},3^{2},4^{2} \right }  $ ?  

  1. $ \displaystyle \left \{ 2^{2},4^{2} \right \} $
  2. $ \displaystyle \left \{ 1^{2},5^{2} \right \} $
  3. $ \displaystyle \left \{ 1^{2},5^{2},6 ^{2} \right \} $
  4. $ \displaystyle \left \{ 1^{2},3^{2},5^{2},6^{2} \right \} $

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

$A\cap B={2^2,4^2}$
$\bar{A\cap B}=U-(A\cap B)={1^2,3^2,5^2,6^2}$
Option D is correct.