Tag: basic operations on sets

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

A survey shows that $63\%$ of Indians like mangoes whereas $76\%$ like apple. If $x%$ of the Indians like both mangoes and apples, then

  1. $x=39$
  2. $x=63$
  3. $39\le x\le 63$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation
Let $100$ denote the population of India, then
$n(A)=63$, $n(B)=76$, $n(A\cap B)=x$
$n(A\cup B)=n(A)+n(B)-n(A\cap B)$
$\therefore x=63+76-n(A\cap B)$ ..$(1)$
If should be noted that $n(A\cup B)\neq 100$ but $n(A\cup B)\le 100$ because there are Indians who may like other fruits besides mangoes and apples.
Hence from $(1)$, $x=139-n(A\cup B)$
$\therefore x\ge 39$ or $39\le x$
Again $A\cap B\subset A$, $A\cap B\subset B$
$\therefore n(A\cap B)\le n(A)=63$, $n(A\cap B)\le n(B)=76$
$\therefore x\le 63$.
Hence $39\le x\le 63$.
Multiple choice mathematics and statistics introduction to set union and intersections union and intersection of sets basic operations on sets

Two set A and B are as under 
A = {(a,b) $\epsilon$ R $\times$ R : $\mid a - 5\mid$ < $1$ and  $\mid b - 5\mid$ < $1$};
B = {(a,b) $\epsilon$ R $\times$ R : $4(a-6)^2 + 9(b-5)^2$ $\leq 36$. Then, 

  1. B $\subset$ A
  2. A $\subset$ B
  3. A $\bigcap$ B = $\phi$ (an empty set)
  4. nither A $\subset$ B nor B $\subset$ A
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Set A is a square centered at (5, 5) with side length 2. Set B is an ellipse centered at (6, 5) with semi-axes 3 and 2. By comparing the boundaries, one can show that all points in A satisfy the inequality for B.

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

$R$ is the set of all positive odd integers less than $20$; $S$ is the set of all multiples of $3$ that are less than $20$. How many elements are in the set $R$ $\cap$ $S$?

  1. 0

  2. 1

  3. 2

  4. 3

  5. 4
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

Given, $R=$ {$1,3,5,7,9,11,13,15,17,19$} , $S=$ {$3,6,9,12,15,18$}
Therefore the intersection of $S$ and $R$ is {$3,9,15$}.
So the number of elements which are common to both is $3$.

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

Let $Z$ denotes the set of all integers and $A=\left{ \left( a,b \right) :{ a }^{ 2 }+3{ b }^{ 2 }=28,a,b\in Z \right} $ and $B=\left{ \left( a,b \right) :a < b,a,b\in Z \right} $. Then, the number of elements in $A\cap B$ is

  1. $2$
  2. $4$
  3. $6$
  4. $5$
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

$\because A=\left{ \left( a,b \right) :{ a }^{ 2 }+3{ b }^{ 2 }=28,a,b\in Z \right} $
   $=\left{ \left( 5,1 \right) ,\left( -5,-1 \right) ,\left( 5,-1 \right) ,\left( -5,1 \right) ,\left( 4,2 \right) ,\left( -4,-2 \right) ,\left( 4,-2 \right) ,\left( -4,2 \right) ,\left( 1,3 \right) ,\left( -1,-3 \right) ,\left( 1,-3 \right) ,\left( -1,3 \right)  \right}$
and $B=\left{ \left( a,b \right) :a<b,a,b\in Z \right} $
$\therefore A\cap B=\left{ \left( 1,3 \right) ,\left( -1,3 \right) ,\left( -4,-2 \right) ,\left( -4,2 \right) ,\left( -5,-1 \right) ,\left( -5,1 \right)  \right} $
$\therefore $ The number of elements in $A\cap B$ is $6$.

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

If $A={\theta :\tan \theta -\tan^2\theta > 0}, B={\theta :|\sin \theta | < 1/2}$ find $A\cap B$.

  1. $\left(0 , \cfrac{7\pi}{6}\right)$
  2. $\left(0 , \cfrac{\pi}{6}\right)$
  3. $\left(0 , -\cfrac{\pi}{6}\right)$

  4. None of these
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

$\tan \theta - \tan ^2 \theta > 0$ is only true when 

$0<\tan \theta < 1$
or,$ 0< \theta < \cfrac{\pi}{4}$
$A = (0,\cfrac{\pi}{4})$
$|\sin \theta| < \cfrac{1}{2}$
$-\cfrac{1}{2} < \sin \theta < \cfrac{1}{2}$
$B = (-\cfrac{\pi}{6} , \cfrac{\pi}{6})$
$A \cap B = (0 , \cfrac{\pi}{6})$