Sets - Class 11 Commerce Maths - Extra Questions

In an examination, $$35$$% students failed in Hindi, $$45$$% students failed in English and $$20$$% students failed in both the subjects. What is the percentage of students passing in both the subjects?  

P(fail in hindi)$$=P(A)=35$$%
P(fail in english)$$=P(B)=45$$%
P(fail in both)$$=P(A\cap B)=20$$%
$$P(A\cup B)=P(A)+P(B)-P(A\cap B)\\ =35+45-20$$
$$ =60$$% of students failed in atleast one.
No. of students who passed in both 
$$=100-60=40$$%


If $$X$$ and $$Y$$ are two sets such that $$n ( X ) = 17$$,$$n ( Y ) = 23$$ $$n ( X \cup Y ) = 38$$ find $$n ( X \cap Y )$$

$$n ( X ) = 17$$
$$n ( Y ) = 23$$ 
$$n ( X \cup Y ) =38$$
$$n ( X \cap Y )=n(X)+n(Y)-n ( X \cup Y )$$
$$n ( X \cap Y )=17+23-38=2$$
$$n ( X \cap Y )=2$$



Let $$O=$$ Set of odd natural numbers $$=\{1,3,5,7,9,...\}$$ and $$E=$$Set of even natural numbers $$=\{2,4,6,8,10,........\}$$. Then show that $$28\in E$$.

$$0 =  \{1,3,5,7,9,.....\} $$
$$E = \{2,4,6,8,10 ,...... \}$$
It is clear from the definition of $$E$$, $$28  \in  \,E\,$$


If $$A=\left\{1, 3, 9, 11, 13\right\}$$ and $$B=\left\{1, 9, 11\right\}$$ then find $$A\cup B$$ and $$A\cap B$$.

Given,
 $$A=\left\{1, 3, 9, 11, 13\right\}$$ and $$B=\left\{1, 9, 11\right\}$$.
Now,
$$A\cup B$$
$$=\{1,3,9,11,13\}$$.
And 
$$A\cap B$$
$$=\{1,9,11\}$$.


In a group of $$950$$ persons, $$750$$ can speak Hindi and $$460$$ can speak English. Find how many can speak both Hindi and English.

Let A and B denote the sets of persons who can speak Hindi and English respectively. 

Then $$n(A\cup B)=950,$$ 

$$ n(A)=750$$ 

$$n(B)=460$$.

$$n(A\cap B)=n(A)+n(B)-n(A\cup B)$$ 

$$750+460-950=260$$.

260 persons can speak both Hindi and English.


Let $$u=\{1, 2, 3, 4, 5, 6, 7, 8, 9\}, A=\{1, 2, 3, 4\}, B=\{2, 4, 6, 8\}$$ and $$C=\{3, 4, 5, 6\}$$. Find B'.

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

$$ A=\{1, 2, 3, 4\}$$ 

$$ B=\{2, 4, 6, 8\}$$ 

$$C=\{3, 4, 5, 6\}$$

$$B^{\prime}=u-B$$ = $$\{1,2,3,4,5,6,7,8,9\}-\{2,4,6,8\}$$

= $$\{1, 3, 5, 7, 9\}$$.


If $$A -B = \phi$$ then $$A \cup B$$ =............

Given,
$$A-B=\phi$$.
Then 
$$A\cup B$$
$$=(A-B)\cup(A\cap B)\cup (B-A)$$
$$=(A\cup B)\cup (B-A)$$ [ Using (1)]
$$=B$$.


Complete the following statements 
1.A∩B=B⟹B........AA∩B=B⟹B........A

1. $$A\bigcap { B } =B$$
It means that A is subset of B $$\left( A\subset B \right) $$
$$B\supset A$$ (B is a superset of A)
2. $$A\subset B$$
(Image 1)
$$n\left( A-B \right) $$
$$A-B=\phi \left( \because A\subset B \right) $$
$$n\left( A-B \right) =0$$
3. $$A\subset B$$
(Image 2)
$$A\bigcap { \left( A\bigcup { B }  \right)  } '$$
$$A\bigcup { B } =B\left( \because A\subset B \right) $$
$$A\bigcap { B' } $$
$$=\phi $$ (From the diagram)
4. $$A-B=\phi $$
(Image 3)
$$n\left( A\bigcup { B }  \right) $$
$$=A+B$$
$$n\left( A\bigcup { B }  \right) =n\left( A \right) +n\left( B \right) $$

1177936_1358276_ans_ae3e5de06871491d8a7a5953f130039e.png


Solve :- $$-12x>30,$$ when 
(i) $$x$$ is a natural number 
(ii) $$x$$ is an integer.

$$-12x>30$$
or, $$12x<-30$$
or, $$x<\cfrac { -5 }{ 2 } $$
i) When $$x\in \mathbb{N}\quad x=\phi $$
ii) When $$x\epsilon I\quad x\in \left[ -\infty ,-3 \right] $$


If $$A=\{1, 2\}$$ and $$B=\{1, 3\}$$, find $$A\times B$$ and $$B\times A$$.


1421533_1666122_ans_12521a9695334e7b93f0d507790204d2.jpg


If $$A=\{1, 2, 3\}$$ and $$B=\{2, 4\}$$, what are $$A\times B, B\times A, A\times A, B\times B$$, and $$(A\times B)\cap (B\times A)$$


1421543_1666134_ans_22c7a38dca5d499c86c9d3ed58ef608d.jpg


Let $$A=\{1, 2, 3\}$$ and $$B=\{3, 4\}$$. Find $$A\times B$$ and show it graphically.


1421537_1666125_ans_cb30e45ca6614ba8b3de9e4f095823ac.jpg


If A and B are two sets having $$3$$ elements in common. If $$n(A)=5, n(B)=4$$, find $$n(A\times B)$$ and $$n\{(A\times B)\cap (B\times A)\}$$.


1421609_1666142_ans_a134db98f3464561a2739fa3e768a908.jpg


Class 11 Commerce Maths Extra Questions