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 language different sets de morgan's law de morgan's law for set theory

Let $A$ and $B$ are two finite sets such that $n(A)=3$ and $n(B)=4$ then  the number of elements in $A\Delta B$.

  1. $2$
  2. $7$
  3. $5$

  4. can not be determined
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Now, we have,

$A\Delta B=(A-B)\cup(B-A)$.
But it is impossible to find the number of elements in the set $A\Delta B$ as the sets $A$ and $B$ are not given explicitly. 

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

$A\cup B=A\cap B$ if and only if

  1. A is an empty set

  2. B is an empty set

  3. Both A and B are empty sets

  4. Both A and B are non-empty sets
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Solution:- Lets assume A is an empty set and B=$\left{ a,b \right}$

Now $A\cup B=\left{ a,b \right}$  and $A\cap B=\oslash $, so in all cases other than C , the condition is not satisfied. So C is the correct answer.

Multiple choice business maths functions and graphs some functions and their graphs -i graphs of the form y=ax^2+bx+c introduction to sets

The number of elements of an identity function defined on a set containing four elements is______

  1. $\displaystyle 2^{2}$
  2. $\displaystyle 2^{4}$
  3. $\displaystyle 2^{8}$
  4. $\displaystyle 2^{16}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

If an element is related to itself, it is called an identity function. That is $ f(x) = x $

So, if  the set has $ 4 $ elements, then the function will also have $ 4 = 2^2 $ elements.

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

A and B are two sets such that $A\displaystyle\cup B$ has $18$ elements If A has $8$ elements and B has $15$ elements then the number of elements in $A\displaystyle\cap  B$ will be: 

  1. $5$
  2. $8$
  3. $7$
  4. $4$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

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

$n(A \cap B) = n(A)+n(B)-n(A \cup B)=8+15-18=5$

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

Let $A = { x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 2$ }$
     $ B = { x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 5$}$
     $C = {x | x$ $\displaystyle \in $ $N$, $x$ is a multiple of 10$}$
The set $\displaystyle\left ( A\cap B \right )\cap C$ is equal to:

  1. $A$
  2. $\displaystyle A \cap C$
  3. $B$
  4. $C$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation
Given A = $\{ 2, 4, 6, 8, 10, 12, 14,...\}$  
B = $\{5, 10, 15, 20, 25,...\}$
C = $\{10, 20, 30, 40, ...\}$
$\displaystyle \Rightarrow $$\displaystyle A\cap B$ = $\{ 10, 20, 30, ...\}$ 
($\displaystyle A\cap B$) $\displaystyle \cap C=$ $\{10, 20, 30, ...\}$ = C
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Given $A={a,b,c,d,e,f,g,h}$ and $B={a,e,i,o,u}$ then $A\cap B$ is equal to

  1. $\{a,e\}$
  2. $\{f,g\}$
  3. $\{g,h\}$
  4. $\{i,u\}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation
Given, $A=\{a,b,c,d,e,f,g,h\}$ and $B=\{a,e,i,o,u\}$

$ A$ intersection $B $, which means a new set can be constructed by determining which members are common among the two sets.

So as per the question:-
$A\cap B=\left\{ a,e \right\}$

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

If $A = \left {1, 2, 3, 4, 5, 6, 7, 8\right }$ and $B \left {1, 3, 5, 7\right }$, then find $A - B$ and $A \cap B$

  1. $\left \{3, 5\right \}$ and $\left \{2, 4, 6\right \}$
  2. $\left \{2, 4, 6\right \}$ and $\left \{1, 5\right \}$
  3. $\left \{2, 4, 6, 8\right \}$ and $\left \{1, 3, 5, 7\right \}$
  4. $\left \{1, 3, 5, 8\right \}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$A=\{1,2,3,4,5,6,7,8\}$

$B=\{1,3,5,7\}$

$A-B=\{1,2,3,4,5,6,7,8\} - \{1,3,5,7\} = \{2,4,6,8\}$

$A \cap B = \{1,2,3,4,5,6,7,8\} \cap \{1,3,5,7\} =\{1,3,5,7\}$

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

If x belongs to set of integers, A is the solution set of $2(x-1)< 3x-1$ and B is the solution set of $4x-3\leq 8+x$, find A$\cap$B.

  1. $\left\{0, 1, 2\right\}$
  2. $\left\{1, 2, 3\right\}$
  3. $\left\{0, 1, 2, 3\right\}$
  4. $\left\{0, 2, 4\right\}$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
$2(x-1)< 3x-1$
$\Rightarrow 2x-2< 3x-1$
$\Rightarrow 2x-3x< -1+2$
$\Rightarrow -x< 1$
$\Rightarrow x> -1$

$4x-3\leq 8+x$
$4x-3-x+3\leq 8+x-x+3$
$\Rightarrow 4x-x\leq 8+3$
$\Rightarrow 3x\leq 11$
$x\leq \dfrac{11}{3}\Rightarrow x\le 3.67$
Then $-1< x\leq 3.67$
Then, $A\cap B$ is $ \{0,1,2,3 \}$
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

If $A = \left {2, 3, 4, 8, 10\right }, B = \left (3, 4, 5, 10, 12\right }, C = \left {4, 5, 6, 12, 14\right }$, then $(A\cap B)\cup (A\cap C)$ is equal to

  1. $\left \{3, 4, 10\right \}$
  2. $\left \{2, 9, 10\right \}$
  3. $\left \{4, 5, 6\right \}$
  4. $\left \{3, 5, 14\right \}$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

$ A={2,3,4,8,10} , B={3,4,5,10,12} , C={4,5,6,12,14}$


To find : $(A\cap B)$ $\cup  (A\cap C)$

$A\cap B$ = elements common in sets $A$ and $B$. 
$A\cap C$ = elements common in sets $A$ and $C$ 

$A\cap C = \{4\}$ 
$(A\cap B)\cup (A\cap C)=$elements in $(A\cap B)$ and $(A\cap C)$
$(A\cap B) \cup  (A\cap C) = \{3,4,10\} \cup \{4\} $
$(A\cap B) \cup  (A\cap C)={3,4,10}$

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

Which is the simplified representation of 
$\left( {{A^/} \cap \,{B^/} \cap \,C} \right) \cup \left( {B\, \cap \,C} \right) \cup \left( {A \cap \,C} \right)$  

where A,B,C are subsets of X

  1. A

  2. B

  3. C
  4. $X \cap \,\left( {A \cup B \cup C} \right)$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

Using set theory laws, simplify the expression ((A' intersect B') intersect C) union (B intersect C) union (A intersect C). Notice that C is a common factor across the union terms: (A' intersect B') intersect C can be written as (A union B)' intersect C. Distributing C across the union with other terms using set identities yields simply C.

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