Tag: complement of sets

Questions Related to complement of sets

If among natural numbers $A={5,6,7}$ and $B={8,9,10}$ , then 

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $A \cap B =$ null

  2. $(A$ $\cup$ $B)' = A'\cap B'$

  3. $A$ $\cap$ $B = {2,3,4}$

  4. None of these

Show answer
Correct Option: A,B
Explanation:

Since, no elements of A and B are the same.
Therefore, the Intersection is empty.
i.e. $A \cap B = $ null
We know, $U = $  { $ N $ }
$A = $  { $ 5,6,7 $ }
$B = $  { $ 8,9,10 $ } 
($A \cup B$) $=$ {$5,6,7$) $ \cup $ {$8,9,10$}
               $=$ {$5,6,7,8,9,10$}
Therefore,  ($A \cup B$)'$= $ All natural numbers except $ {5,6,7,8,9,10} \dots (i)$
Now, $A'= $ All Natural numbers except ${5,6,7}$
and $B'=$ All Natural numbers except ${8,9,10}$
Therefore, $A' \cap B' = $ All Natural numbers except ${5,6,7,8,9,10}\dots (ii)$
By comparing $(i) $ and $(ii)$ we get,
$(A \cup B) = A' \cap B'$

Given $A={x\in N :x<6} ,B={3,6,9}$ and $C={x \in N: 2x-5\le 8}$

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $A \cup $ (B $ \cap $C)=(A $ \cap  $B) $\cap $(A $ \cap$ C)

  2. (A $\cup$B)'=A'$\cap$B'

  3. A $\cup$B=null set

  4. None of the above

Show answer
Correct Option: B
Explanation:

$A = { x \in N : x < 6 }$

$\therefore A = { 1,2,3,4,5 }$
$B = { 3,6,9 }$
$C = { x \in N : 2x - 5 \leq 8 }$
$\Rightarrow C = { 1,2,3,4,5,6 }$
Option A. $A \cup (B \cap C) = A \cup {3, 6 } = { 1,2,3,4,5,6 }$
RHS $= (A \cap B) \cap ( A \cup C) = { 3 } \cap {1,2,3,4,5,6} = {3}$
LHS $\neq$ RHS

Option B. $(A \cup B)' = A' \cap B'$ is always true by De Morgan's Law  . 
$A\cup B={1,2,3,4,5,6,9}$
$(A\cup B)'={7,8,10,11,...}$
$A'={x\in N:x\geq 7}$ and $B'={1,2,4,5,7,8,10,11,12...}$
$A'\cap B'={7,8,10,11,12...}$
$\therefore (A\cup B)'=A'\cap B'$

Option C. $A \cap B = { 3 } \neq$ null set
Hence, only B is correct.

If $A, B$ be any two sets, then $(A\cup B)'$ is equal to

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $A'\cup B'$

  2. $A'\cap B'$

  3. $A\cap B$

  4. $A\cup B$

Show answer
Correct Option: B
Explanation:

It is De Morgan's law that $(A\cup B)' = A'\cap B'$

If $U = {3, 4, 5, 6, 7, 8, 9}, X = {3, 4}, Y = {5, 6}$ and $Z = {7, 8, 9}$, then $\displaystyle Y'\cap \left ( X\cap Z \right )'$ is equal to 

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $\displaystyle X\cup Y$

  2. $\displaystyle Y\cup Z$

  3. $\displaystyle X'\cap Y'$

  4. $X\cup Z$

Show answer
Correct Option: D
Explanation:

$ { Y ' } = U - Y = { 3, 4, 7, 8, 9} $

$ X\cap Z = {0} $ as there is no element common in sets X and Z

$\Rightarrow (X\cap Z) ' = U - (X\cap Z) =  {3, 4, 5, 6, 7, 8, 9} $

Now, $Y' \cap (X\cap Z)'  = { 3, 4, 7, 8, 9} \cap {3, 4, 5, 6, 7, 8, 9} ={ 3, 4, 7, 8, 9} $

This is equal to $ X \cup Z = {3,4,7,8,9} $

If $U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$, $A = {0, 3, 4, 7}$ ,$B = {1, 2, 8, 9}$
then $(A U B)'$ is

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. ${2, 5}$

  2. ${5, 6}$

  3. ${8, 9}$

  4. ${6, 7}$

Show answer
Correct Option: B
Explanation:

Given, $U = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}$, $A = {0, 3, 4, 7}$ ,$B = {1, 2, 8, 9}$


$A^c={1,2,5,6,8,9}, B^c= {0,3,4,5,6,7} $

By De Morgan's law

$(A\cup B)^c = A^c\cap B^c$

$=\{1,2,5,6,8,9\}\cap \{0,3,4,5,6,7\}$

$=\{5,6\}$

Out of 800 boys in a school 224 played cricket, 240 played hockey and 236 played basketball. Of the total 64 played both basketball and hockey, 80 played cricket and basketball and 40 played cricket and hockey, 24 players all the three games. The number of boys who did not play any game is

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $128$

  2. $216$

  3. $240$

  4. $260$

Show answer
Correct Option: D
Explanation:

No. of players who played at least one game is:


By set theory

$n(C\cup H\cup B)= n(C) +n(H) +n(B)-n(B\cap H)-n(C\cap B) -n(C\cap H) + n(C\cap H \cap B)$

$=224+240+236-64-80-40+24=540$

Hence $260$ players do not play any game.

Find the De Morgan's law of intersection.

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $(A\cap B)^{'} = A \cup B^{'}$

  2. $(A\cap B)^{'} = A^{'} \cup B^{'}$

  3. $(A\cup B)^{'} = A^{'} \cup B^{'}$

  4. $(A\cap B)^{'} = A^{'} \cap B^{'}$

Show answer
Correct Option: B
Explanation:

De Morgan's law of intersection is $(A\cap B )' = A' \cup B' $

Find the De Morgan's law of union.

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $(A\cap B)^{'} = A^{'} \cap B^{'}$

  2. $(A\cup B)^{'} = A^{'} \cap B^{'}$

  3. $(A\cup B)^{'} = A^{'} \cup B^{'}$

  4. $(A\cup B)^{'} = A\cap B^{'}$

Show answer
Correct Option: B
Explanation:

De Morgan's law of union is 

$(A\cap B)' = A'\cap B'$

For any two sets A and B, $\left { (A\setminus B)\cup (B\setminus A) \right }\cap (A\cap B)$ is:

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $\phi $

  2. $ A \cup B$

  3. $A \cap B$

  4. $A' \cap B'$

Show answer
Correct Option: A
Explanation:

$(A$ \ $B)\cup(B$ \ $A)$ contains that elements form set $A$ and $B$ that are not contained in the other set.

$(A$ \ $B)\cup(B$ \ $A)=(A\cap B)$ \ $(A\cap B)$
$\therefore (A$ \ $B)\cup(B$ \ $A)$ does not contain elements of $A\cap B$
Hence {$(A$ \ $B)\cup(B$ \ $A)$}$\cap (A\cap B)=${$\phi$}

In order to draw a graph of $f(x) = ax^{2} + bx + c$, a table of values was constructed. These values of the function for a set of equally spaced increasing values of $x$ were $3844, 4096, 4227, 4356, 4489, 4624$, and $4761$. The one which is incorrect is

mathematics and statistics set language de morgan's law for set theory complement of sets different sets de morgan's law

  1. $4096$

  2. $4356$

  3. $4489$

  4. $4761$

  5. None of these

Show answer
Correct Option: E
Explanation:

We are told that the values of $f(x)$ listed correspond to
7$f(x), f(x + h), f(x + 2h), ...., f(x + 7h)$.
Observe that the difference between successive values is given by
$f(x + h) - f(x) = a(x + h)^{2} + b(x + h) + c - (ax^{2} + bx + c)$
$= 2ahx + ah^{2} + bh$.
Since the difference is a linear function of $x$, it must change by the same amount whenever $x$ is increased by $h$. But the successive differences of the listed values
$3844\ 3969\ 4096\ 4227\ 4356\ 4489\ 4624\ 4761$
are $125\ 127\ 131\ 129\ 133\ 135\ 137$
so that, if only one value is incorrect; $4227$ is the value.