Tag: union and intersection of sets

Questions Related to union and intersection of sets

Let $S=\left{ \left( x,y \right) :\dfrac { y\left( 3x-1 \right)  }{ x\left( 3x-2 \right)  } <0 \right}$ and $S'=\left{ \left( x,y \right) \in A\times B;\ -1\le A\le 1,-1\le B\le 1 \right} $ There area of $S\cap S'$ is

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

  1. $1$

  2. $3$

  3. $2$

  4. $4$

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Correct Option: A

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

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

  1. 13

  2. 16

  3. 9

  4. 10

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Correct Option: A

Let $P={ \theta :sin\theta -cos\theta =\sqrt { 2 } cos\theta } $ and $Q={ sin\theta + cos\theta =\sqrt { 2 } sin\theta } $ be two sets. Then:

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

  1. $P\subset Q\quad and\quad Q-P\neq \emptyset $

  2. $Q\subset P$

  3. $P\subset Q$

  4. $P=Q$

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Correct Option: D
Explanation:

$P = \left{ {\theta :\sin \theta  - \cos \theta  = \sqrt 2 \cos \theta } \right}$

$Q = \left{ {\theta :\sin \theta  + \cos \theta  = \sqrt 2 \sin \theta } \right}$
From $P$
$\sin \theta  - \cos \theta  = \sqrt 2 \cos \theta $
$\sin \theta  = \left( {\sqrt 2  + 1} \right)\cos \theta $
$\frac{{\sin \theta }}{{\cos \theta }} = \left( {\sqrt 2  + 1} \right)$
$\tan \theta  = \left( {\sqrt 2  + 1} \right)$
from $Q$
$\sin \theta  + \cos \theta  = \sqrt 2 \sin \theta $
$\sin \theta \left( {\sqrt 2  - 1} \right) = \cos \theta $
$\frac{{\sin \theta }}{{\cos \theta }} = \left( {\sqrt 2  - 1} \right)$
$\tan \theta  = \left( {\sqrt 2  - 1} \right)$
$\tan \theta  = \frac{1}{{\sqrt 2  - 1}} \times \frac{{\sqrt 2  + 1}}{{\sqrt 2  + 1}} = \sqrt 2  + 1$
$\therefore P = Q$
Hence,
option $(D)$ is correct answer.

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

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

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

  1. True

  2. False

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Correct Option: A
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.

Let $A = {$ multiples of $3$ less than $20 }$
      $B = {$ multiples of $5$ less than $20}$
Then  $A$ $\displaystyle\cap$ $B$ is

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

  1. ${3, 5}$

  2. ${5, 9}$

  3. ${15}$

  4. $\displaystyle\phi $

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Correct Option: C
Explanation:

$A = {$ multiples of $ 3 $ less than $20}$

    $= {3,6,9,12,15,18}$
$B={ $ multiples of $5$ less than $20}$
    $= {5,10,15}$

$A \cap B = 15$

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?

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

  1. $P _1\cap P _{23}$

  2. $P _7\cap P _{21}$

  3. $P _{12}\cap P _{20}$

  4. $P _{20}\cap P _{24}$

Show answer
Correct Option: C
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.

If $A={a,b,c,d,e,f}$ and $B={c,e,f,g,h}$, then the number of elements of $(A-B)\cap A$ is

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

  1. $0$

  2. $1$

  3. $2$

  4. $3$

Show answer
Correct Option: D
Explanation:

$A-B={a,b,d}$
$(A-B)\cap A={a,b,d}$

In a group of $760$ persons, $510$ can speak Hindi and $360$ can speak English. Find how many can speak Hindi only.

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

  1. $250$

  2. $400$

  3. $1270$

  4. $150$

Show answer
Correct Option: B
Explanation:
People who can speak both Hindi and English $= n (H ∩ E)$

$n (P) = n(E) + n(H) – n (H ∩ E)$

$n(E\cap H)=510+360-760=110$

We can see that, $H$ is disjoint union of $n(H–E)$ and $n (H ∩ E).$


(If $A$ and $B$ are disjoint then $n (A ∪ B) = n(A) + n(B))$

$∴ H = n(H–E) ∪ n (H ∩ E).$

$⇒ n(H) = n(H–E) + n (H ∩ E).$

$⇒ 510 = n (H – E)+ 110$

$⇒ n(H–E) = 400.$

There are $25$ trays on a table in the cafeteria. Each tray contains a cup only, a plate only, or both a cup and a plate. If $15$ of the trays contain cups and $21$ of the trays contain plates, how many contain both a cup and a plate?

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

  1. $10$

  2. $11$

  3. $12$

  4. $13$

Show answer
Correct Option: B
Explanation:
Let $'A'$ denote the set with a cup
$'B'$ denote the set with a plate
$'A$ $\cap$ $B'$ denote the set with both a cup and a plate
$'A$ $\cup$ $B'$ denote the set with the trays either a cup or a plate
$n($ $X)$ denote the number of elements in the set $'X'$

Given, total number of trays $n(A$ $\cup$ $B)$ $=$ $25$
Number of trays that contain cups $n($$A)$ $=$ $15$
Number of trays that contain cups $n($$B)$ $=$ $21$

To find the trays with both a cup and a plate $n(A$ $\cap$ $B)$,

We know that
$n(A$ $\cup$ $B)$ $=$ $n($$A)$$+$ $n($$B)$ $-$ $n(A$ $\cap$ $B)$
Rearranging the terms, we get
$n(A$ $\cap$ $B)$ $=$ $n($$A)$$+$ $n($$B)$ $-$ $n(A$ $\cup$ $B)$
From the above,
$n(A$ $\cap$ $B)$ $=$ $15$ $+$ $21$ $-$ $25$
$=$ $11$
$n(A$ $\cap$ $B)$ $=$ $11$

Therefore, number of trays with both a cup and a plate is $'11'$.

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$ ?

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

  1. 4

  2. 5

  3. 9

  4. None of the above

Show answer
Correct Option: A
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, $n(a \cap B) = 4$.