Tag: different sets

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Questions Related to different sets

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

In a battle $70\% $ of the combatants lost one eye, $80\% $ an ear, $75\% $ an arm, $85\% $ a leg and $x\% $ lost all the four limbs the minimum value of $x$ is 

  1. $10$
  2. $12$
  3. $15$
  4. $none\ of\ these$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Let the total number of combatants be 100. The number of soldiers who did NOT lose an eye is 30, an ear is 20, an arm is 25, and a leg is 15. The maximum number of soldiers who lost none of these is the sum of those escaping each injury, which is 30 + 20 + 25 + 15 = 90. Therefore, at least 100 - 90 = 10 percent lost all four limbs.

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

If A and B be two sets such that n(A) = 15, n(B) =25, then number of possible values of $n(A\Delta B)$(symmetric difference of  A and B) is

  1. 30

  2. 16

  3. 26

  4. 40
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation
$n(A \triangle B)= n (A \cup B)- n (A \cap B)$
for $n$ (A \triangle B)$ to be max. $n (A \cap B)=0$
We know, that 
$n(A \cup B)= n (A)+ n (B)- n (A \cap B) = 15+25-0=40$
$\Rightarrow n (A \triangle B)_{max} = 40-0 =40$
For minimum value of $n (A \triangle B)$
$n (A \cup B)$ should be min, $n (A \cap B)$ should be max.
$n (A \triangle B)$ min $=25-15= 10$
So. value of 
$n (A \triangle  B)= n (A \cup B)- n(A \cap B)$ lies om the set
${10,11,12,......, 3,9,40}$
Now, when $n (A \triangle B)$ is max. i.e. when 
$n( A \cup B )=40$ & $n (A \cap B)=0$
If we decrease $n (A \cup B)$ by $1$ then $n (A \cap B)$
Will increase by $1$
$n (A \triangle B)=39-1= 38$
Similarly on for the decrease of $1$ you will get in $(A \triangle B)$ as $36$ and $30$ so on.
Hence 
Range of $n (A \triangle B)= {10,12,14,16,18,20,......,38,40}$ 
$=16$ values 
Multiple choice maths set language different sets de morgan's law de morgan's law for set theory

If $A=\left {x\epsilon C: x^2=1\right }$ and $B=\left {x\epsilon C: x^4=1\right }$, then $A\Delta B$ is equal to

  1. $\left \{-1, 1\right \}$
  2. $\left \{-1, 1, i, -i\right \}$
  3. $\left \{-i, i\right \}$

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

$x^2=1\Rightarrow x=-1, 1.\therefore A=\left {-1, 1\right }$
$x^4=1\Rightarrow x^2=-1, 1$
$\Rightarrow x=-i, i, -1, 1.\therefore B=\left {-i, i, -1, 1\right }$
$\therefore A\Delta B=(A-B)\cup (B-A)=\phi \cup \left {-i, i\right }=\left {-i, i\right }$.