Tag: introduction to set

Questions Related to introduction to set

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

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

  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

Show answer
Correct Option: B
Explanation:

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

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

The number of subsets of the set $A={ { a } _{ 1 },{ a } _{ 2 },.........{ a } _{ n }} $ which contain even number of elements is

maths introduction to set intervals subsets subsets and supersets

  1. ${ 2 }^{ n-1 }$

  2. ${ 2 }^{ n }-1$

  3. ${ 2 }^{ n }-2$

  4. ${ 2 }^{ n }$

Show answer
Correct Option: A
Explanation:
The total no of subsets of $A$ is the cardinality of the power set of $A$. 

So, if $|A|=n$ then $|P(A)|=2^n$. 

Therefore total no of subsets of $A$ is $2^n$.

Similarly,

The even number of events is given by, $2^{n-1}$