Tag: de morgan's law for set theory

Questions Related to de morgan's law for set theory

$|x|$ represent number of elements in region X. Now the following conditions are given
$|U|=14$, $|(A-B)^C|=12$, $|A\cup B|=9$ and $|A\Delta B|=7$, where A and B are two subsets of the universal set U and $A^C$ represents complement of set A, then?

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

  1. $|A|=2$

  2. $|B|=5$

  3. $|A|=4$

  4. $|B|=7$

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Correct Option: A

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 

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

  1. $10$

  2. $12$

  3. $15$

  4. $none\ of\ these$

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Correct Option: B

Let $A$ and $B$ are two finite sets such that $n(A)=3$ and $n(B)=4$ then  the number of elements in $A\Delta B$.

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

  1. $2$

  2. $7$

  3. $5$

  4. can not be determined

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Correct Option: D
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. 

$A\cup B=A\cap B$ if and only if

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

  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

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Correct Option: C
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.

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

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

  1. 30

  2. 16

  3. 26

  4. 40

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Correct Option: B
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 

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

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

  1. $\left {-1, 1\right }$

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

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

  4. None of these

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Correct Option: C
Explanation:

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