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 introduction to set cardinal number of a finite set cardinality of a set representation of sets

Let $A$ and $B$ be two sets such that $n(A)=70, n(B)=60$ and $n(A\cup B)=110$. Then $n(A\cap B)$ is equal to-

  1. $240$
  2. $20$
  3. $100$
  4. $120$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

We know,  $n(A\cup B)=n(A)+n(B)-n(A\cap B)$


$\therefore  n(A\cap B)=n(A)+n(B)-n(A\cup B)$

                      $=70+60-110$

                      $=20$

Multiple choice maths introduction to set cardinal number of a finite set cardinality of a set representation of sets

Let the sets $A={2,4,6, 8, ...}$ and $B={3, 6, 9, 12, ...}$, and $n(A)=200, n(B)=250$. Then

  1. $n\left ( A\cap B \right )=67$
  2. $n\left ( A\cup B \right )=450$
  3. $n\left ( A\cap B \right )=66$
  4. $n\left ( A\cup B \right )=384$
Reveal answer Fill a bubble to check yourself
C,D Correct answer
Explanation

In A, last term will be $400$.

In B, the terms are also in A.P having a common difference of $3$.

Hence 

$a _{n}=a _1+(n-1)d$.

Now $n=250$ for the last term.

Hence

$a _{250}=3+(250-1).3$
$=3(1+250-1)$
$=750.$

Now $A\cap B$ will have elements which are multiples of $6$.

Last term will be $400-4=396$.

Hence
$a _{n}=a+(n-1).d$
$d=6,n=?,a=6$ and $a _{n}=396$

Hence
$396=6+(n-1).6$
Or 
$66=n$.

Hence
$n(A\cap B)=66$.

Now 
$n(A \cup B)=n(A)+n(B)-n(A\cap B)$
$=200+250-66$
$=384$.

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

The set of all animals on the earth is a 

  1. Finite set

  2. Singleton set

  3. Null set

  4. Infinite set
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Since the numbers of animals on the earth are countable(limited).
Therefore, set of all the animals on the earth is "finite" set.
Option A is correct.

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

Which of the following sets is non - empty ?

  1. Set of odd natural numbers divisible by 2
  2. ${x: x+4=0, x\in N}$

  3. Set of even prime numbers
  4. ${x:2< x< 3,x\in N}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A non empty set has atleast one element.

From the given options we see that the SET C has one element which is $ 2 $

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

For any three sets A, B and C, $A \cap (B \cup C)$ is

  1. $(A\cup B)\cup (B\cap C)$
  2. $(A\cap B)\cup (A\cap C)$
  3. $A\cup (B\cap C)$
  4. $(A\cup B)\cap (B\cup C)$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

In the problem statement we are taking union of $B$ and $C$ and then taking its intersection with $A$.

This means $A\cap (B\cup C)$will contain elements that are in $A$ and are in either $B$ or $C$.
$\therefore A\cap (B\cup C) $ is equivalent to taking intersection of $A,B$ and $A,C$ and then taking there union i.e. $(A\cap B)\cup(A\cap C)$
Hence, option B is correct.

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

Define finite set.
Is $A=$set of animals on the earth a finite set.

  1. True

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

Definition: A finite set is a set which has fixed or finite number of elements.
$A=$ Set of animals on the earth is a finite set because number of animals on the earth is large in number, but it is finite/fixed.

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

State which of the following are finite sets.
$(i){x:x\in N }$ and $(x-1)(x-2)=0.$
$(ii){x:x\in N }$ and $x$ is prime.
$(ii){x:x\in N }$ and $x$ is odd.

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

$(i)(x-1)(x-2)=0$

$\Rightarrow x-1=0$ or $x-2=0$
$\Rightarrow x=1$ or $x=2$, Number of solutions of equation are finite.
$(ii){2,3,5,7,11,...}$, This set contains infinte number of elements.
$(iii){..,1,3,5,7,...}$, This set contains infinite number of elements. 

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

Which of the following sets is not a finite set ?

  1. $\{ (x,y):{ x }^{ 2 }+{ y }^{ 2 }\le 1\le x+y,\ \ x,y\in R\} $
  2. $\{ (x,y):{ x }^{ 2 }+{ y }^{ 2 }\le 1\le x+y,\ \ x,y\in Z\} $
  3. $\{ (x,y):{ x }^{ 2 }\le y\le |x|,\ \ x,y\in Z\} $
  4. $\{ (x,y):{ x }^{ 2 }+{ y }^{ 2 }=1,\ \ x,y\in Z\} $
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The set ${ (x,y):{ x }^{ 2 }+{ y }^{ 2 }\le 1\le x+y,\ \ x,y\in R} $ consists of all the points in the first quadrant which lie inside the circle ${ x }^{ 2 }+{ y }^{ 2 }=1$ and above the line $x+y=1$ .So, it is not a finite set.
Option $A$ is correct.

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

Which of the following is incorrect.

  1. The power set of an infinite set is infinite.

  2. The union of two infinite set is infinite.

  3. The intersection of two infinite set is infinite.

  4. The cardinality of an infinite set is infinite.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The intersection of an infinite set may be finite.
Example:
$A={x:x\in N; x>2}$
$B={x:x\in I; x<5}$
Here, Both $A$ and $B$ are infinite but its intersection are finite.

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

Define infinite set .
Is ${x:x\in R:1\le x\le 3}$ a infinite set?

  1. True

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

Definition: A set having infinite number of elements is known as infinite set.
 ${x:x\in R:1\le x\le 3}$ is a infinite set because since $x\in R$, there are infinte number of real numbers lie in between two numbers.