Tag: maths

Questions Related to maths

If $n(A) = n(B)$ then

  1. $n(A - B) = n(B - A)$

  2. $n(AB) = n(A) + n(B)$

  3. $n(A - B) =\phi$

  4. $n(AB) = n(B) - n(A - B)$


Correct Option: A,C
Explanation:

If, $n(A)=n(B)$

a.  $n(A-B) = n(B-A)$. As, the no. of elements are same, if we subtract A from B or B from A, we will get the same no. of elements

b.  $n(AB)\neq n(A)+n(B)$. It is $n(A\cup B)=n(A)+n(B)$

c.  $n(A-B)= \phi$

d.  $n(AB)\neq n(B) - n(A-B)$. It is $n(AB) = n^2$ where n is the no. of elements of these sets

If $n(A) = n(B)$ then:

  1. $n(A- B) = n(B- A)$

  2. $n(AB)= n(A) + n(B)$

  3. $n(A- B)=n(A)-n(B)$

  4. $n(AB) = n(B) - n(A-B)$


Correct Option: A
Explanation:
Given: $n(A) = n(B)$
$n(A)$ is the cardinal no. of set $A$ and same for the set $B$

Thus, the number of elements are always same no matter what type of operation we are performing.

Hence, $n(A-B)=n(B-A)$.

The set contains $5$ elements, then the number of elements in the power set $P$ $(A)$ is equal to

  1. $32$

  2. $36$

  3. $25$

  4. $40$


Correct Option: A
Explanation:

$$n\left( A \right) =5$ 

$\Rightarrow n\left( P\left( A \right)  \right) ==2^{n(A)}={ 2 }^{ 5 }=32$

Number of elements in  a set is called __________

  1. Cardial number

  2. Set number

  3. Members

  4. None


Correct Option: A

In a city $20\%$ of the population travels by car, $50\%$ travels by bus and $10\%$ travels by both car and bus. Then, persons travelling by car or bus is

  1. $80\%$

  2. $40\%$

  3. $60\%$

  4. $70\%$


Correct Option: C

The number of elements of the power set of a set containing $n$ elements is

  1. $2^{n-1}$

  2. $2^n$

  3. $2^n-1$

  4. $2^{n+1}$


Correct Option: B
Explanation:

If there is a finite number of n elements in $A,$ then the power set $P (A)$ has $2^n$ elements. 

Hence option $B$ is the correct answer.

Let $U$ be the universal set for sets $A$ and $B$ such that $n(A)=200 , n(B)=300$ and $n(A\cap B)=100$, then $n(A'\cap B')$ is equal to $300$ provided that $n(U)$ is equal to

  1. $600$

  2. $700$

  3. $800$

  4. $900$


Correct Option: B
Explanation:

$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$=200+300-100$
$=400$
$n(A'\cap B')=n(A\cup B)'$
                    $=n(U) - n(A\cup B)$
$300=n(U)-400$
$n(U)=700$

If $\displaystyle n(U)=700,n(A)= 200,n(B)= 240,n(A\cap B)= 100,$ then $\displaystyle n(A'\cup B') $ is equal to

  1. $260$

  2. $560$

  3. $360$

  4. $600$


Correct Option: D
Explanation:

Using De Morgan's law,
$\displaystyle n(A'\cup B')= n(A\cap  B)' $ $\displaystyle

= n(U)-n(A\cap B)=  n(U)-100 $ $\displaystyle = 700-100= 600$

A market research group conducted a survey of $2500$ consumers and reported that $1620$ consumers like product $p _{1}$ and $1500$ consumers like product $p _{2}$ then (Note $A$ and $B$ denotes the set of products $p _{1}$ and $p _{2}$ respectively)

  1. $\displaystyle n\left ( A \cup B \right )\geq 620$

  2. $\displaystyle n\left ( A \cap B \right )\leq 1500$

  3. $\displaystyle 620 \leq n\left ( A \cap B \right )\leq 1500$

  4. All of these


Correct Option: D
Explanation:

$n(A)  =1620, n(B) = 1500$

$\Rightarrow n(A\cap B) \leq min{n(A), n(B)} $

$\Rightarrow n(A\cap B) \leq 1500$

Also $n(A\cup B) \geq  n(A)+n(B) -n(U) =620$

 $n(A\cup B) \geq620$

Hence all options are correct.

In a community it is found that $52$% people like coffee and $73$% like tea. If $x\%$ like both coffee and tea then

  1. $\displaystyle x\geq 25$

  2. $\displaystyle x\leq 52$

  3. $\displaystyle 25\leq x\leq 52 $

  4. all of these


Correct Option: C
Explanation:

Let $A=$ number of people like coffee, $B=$ number of people like tea.


$\therefore \ n(A)=52$%  $ \ n(B) = 73$%


$\displaystyle n\left ( A\cap B \right ) =x$%

Let total people in the community =$100 \displaystyle = n\left ( U \right )$
 
$\displaystyle \therefore n\left ( A\cup B \right )\leq 100$ 

$\displaystyle n\left ( A \right )+n\left ( B \right )-n\left ( A\cap B \right )\leq 100$ 

$\displaystyle 52+73-x\leq 100$ 

$\Rightarrow \displaystyle 125-x\leq 100$ 

$\displaystyle\Rightarrow  \therefore x\geq 25$...(i)

Again $\displaystyle A\cap B\subseteq A$

$\displaystyle \Rightarrow n\left ( A\cap B \right )\leq n\left ( A \right )$ 

$\displaystyle x \leq 52$ ...(ii)

$\displaystyle

\therefore $ By (i) and (ii)$\displaystyle 25\leq x\leq 52$