A man sells a mobile set for Rs. 5994 at a loss of 7.5%. How much money has he paid to buy the mobile. Also find his loss amount.

Mrs. Chaudhari bought a camera for Rs. 6250. If she sold it at 4% loss, how much was her actual loss ? Find the selling price of the camera.

A shopkeeper purchased an electric fan for Rs. 1560 and sold it at 5% profit. Find his actual profit and selling price of the fan.

Solution of 2

	C.P	S.P	Actual profit or loss	profit or loss %
a)	Rs 250	Rs 300	Rs.50	20%
b)	Rs 640	Rs 608	Rs.32	5%
c)	Rs 800	Rs.832	Rs.32	4%(profit)
d)	Rs 250	Rs 468	Rs 52	10%(loss)
e)	Rs 960	Rs 1008	Rs 48	5%(profit)
f)	Rs 600	Rs 558	Rs 42	7%(loss)
g)	Rs 400	Rs 448	Rs 48(profit)	12%
h)	Rs 864	Rs 756	Rs 108(loss)	12.5%
i)	Rs 1200	Rs 1224	Rs 24	2%(profit)
j)	Rs. 980	Rs 931	Rs 49	5%(loss)

If loss is L% and selling price is S.P. What will be the cost price ?

solution:

If profit is P% and selling price is S.P. What will be the cost price ?

solution:

If loss is L% and cost price is C.P. What will be the selling price ?

solution

If profit is P% and cost price is C.P. What will be the selling price ?

Solution :

नेपाल पृथ्वीको कुल क्षेत्रफलको कति प्रतिशत छ ?

Venn-diagrams

Actually Venn is a mathematician surname called Euler john Venn who first introduced a diagram to represent the sets and its subset. So that diagram is called Venn diagram. So Venn diagram is a diagram like a rectangle, circle or oval shapes used for representing sets. Generally Rectangle represent Universal set and Circles or Ovals inside the rectangle shows its subset.

Following are some Venn diagram which represent subset, disjoint and overlapping set       

i)  AÌU (A is a subset of U).

ii)     A and B are disjoint sets. 

         






                             

iii)    A and B are overlapping sets.     

  











Some Information about John Venn

                                                                               

v  A British logician and philosopher                                  

v  Born : 1834 August 4, Hull, London

v  Famous for Venn diagram

v  Degree in Mathematics (1857) at 23 years old.

v  Notable awards : Fellow of the royal society

v  Died: 1923 April 4 at 88 years old

                                                                                                 

Set operation

There are four main set operations. They are:

Ø  Union of two or more sets[È]

Ø  Intersection of two or more sets[Ç]

Ø  Difference of two sets[-]

Ø  Complement of a set[`A, A^c or A¢]

1)      Union of two or more sets

It means the set of all members which belongs to either single set or to all set.

In two set, Let A and B

The member of the union of these two set should belong to either A or B or both and denoted by AÈB.

In three set, Let A, B and C

The member of the Union of these three sets should belong to either A or B or C or All of three and denoted by AÈBÈC.

In this way we can define for n sets. But in exam we write only for two sets.

There will be two conditions

i) When they are overlapping the repeated member should be written only one time

Consider overlapping set A and B where A= {a,b,c} and B={c,d,e}

Then the union of these two sets A and B symbolically AÈB is {a,b,c,d,e}. here c is repeated so c is taken only one time not repeatedly.

ii) When two are disjoint the problem will not arise.

Consider disjoint set A and B where A={a,c,d} and B={b,e,f}

Then the union of these two sets A and B symbolically AÈB is {a,b,c,d,e,f}

This is an example of two sets. Similarly we can make union of more than two sets when they are overlapping or disjoints.

Generalized Notation : AÈB = {x:xÎA or xÎB}

Representation in Venn diagram when they are overlapping and disjoint.

v  When set A and B are disjoint, then AÈB can be represented in this way by shaded region

v  When set A and B are overlapping, then AÈB can be represented in this way by shaded region.

v  When three sets are given i.e. A, B and C and they are disjoint, then AÈBÈC can be represented         in this way by shaded region

v  When three sets are given i.e A, B and C and they are overlapping, then AÈBÈC can be                       represented in this way by shaded region

Intersection of two or more sets:

It means the set of all members common to all sets.

For two sets

So the intersection of two setA and B  is the set of all members that common to both A and B. It is denoted by AÇB

For three sets

The intersection of three sets A,B and C is the Set of all members that common to all A,B and C. It is denoted by AÇBÇC.

In this way we can define for n sets. But in exam we write only for two sets.

Well in the intersection case, the set is always empty in disjoint set. so it is always exercised in overlapping set.

Representation by Venn diagram through shaded region.

v  When two sets are intersecting.

                                                              

   v  When two sets are disjoint.

                                                                    U

Since the element common to them is zero so there is no shaded region for AÇB.

v  When three sets are overlapping.

                                                                                          

                                             

Difference of two sets

The difference of two set A and B, denoted by A-B is the set of the elements of A which do not belong to B.

For example if A= {1,2,3,4,5}, B={1,3,5,7,9} then A-B={2,4} since 1,3,5 are also in B so it is not collected and the element 7, 9 is ignored, so the remaining element 2,4 are the elements of the set A-B. Similarly we get B-A= {7,9}

Generalized Notation: A-B= {x: xÎA, but xÏA} similarly B-A= {x:xÎB, but xÏA}

Representation on Venn diagram

1.      
       Disjoint case

                                                                            A-B                                                                  B-A

2.      Overlapping case

                     A-B                                                               B-A

Complement of a set

If A be a subset of a universal set U, then the compliment of A, denoted by`A or A^c or A¢ is set of the elements of U which do not belong to P.

It is the difference of a two sets where a set from which another set is deducted  is universal set and the another set is its subset.

        i.e.             

`A or Ac or A¢ = U-A

For example:

If U= {1, 2, 3, 4……………….10} and A= {1,4,5,7}

Then`A= U-P

              = {1,2,3,4………….10} – {1,4,5,7}

              = {2,3,6,8,9,10}

Generalized Notation : `A ={x:xÎU, but xÏA}

Representation on Venn diagram

                                       

                  

 

                                 

Practice Test on Operations on Sets

1. If A = {2, 3, 4, 5}     B = {4, 5, 6, 7}     C = {6, 7, 8, 9}     D = {8, 9, 10, 11}, find 

(a) A ∪ B 

(b) A ∪ C 

(c) B ∪ C 

(d) B ∪ D 

(e) (A ∪ B) ∪ C

(f) A ∪ (B ∪ C) 

(g) B ∪ (C ∪ D) 


2. If A = {4, 6, 8, 10, 12} B = {8, 10, 12, 14} C = {12, 14, 16} D = {16, 18}, find

(a) A ∩ B

(b) B ∩ C

(c) A ∩ (C ∩ D) 

(d) A ∩ C

(e) B ∩ D

(f)(A ∩ B) ∪ C 

(g) A ∩ (B ∪ D) 

(h) (A ∩ B) ∪ (B ∩ C) 

(i) (A ∪ D) ∩ (B ∪ C) 


3. If A = {4, 7, 10, 13, 16, 19, 22}   B = {5, 9, 13, 17, 20}
C = {3, 5, 7, 9, 11, 13, 15, 17}   D = {6, 11, 16, 21} then find 

(a) A - C 

(b) D - A

(c) D - B

(d) A - D 

(e) B - C

(f) C - D 

(g) B - A 

(h) B - D 

(i) D - C 

(j) A - B

(k) C - B

(l) C - A

More Practice Test on Operations on Sets
4. If A and B are two sets such that A ⊂ B, then what is A∪B? 

5. Find the union, intersection and the difference (A - B) of the following pairs of sets. 

(a) A = The set of all letters of the word FEAST 

     B = The set of all letters of the word TASTE 

(b) A = {x : x ∈ W, 0 < x ≤ 7} 

     B = {x : x ∈ W, 4 < x < 9} 

(c) A = {x | x ∈ N, x is a factor of 12} 

     B = {x | x ∈ N, x is a multiple of 2, x < 12} 

(d) A = The set of all even numbers less than 12 

     B = The set of all odd numbers less than 11 

(e) A = {x : x ∈ I, -2 < x < 2} 

     B = {x : x ∈ I, -1 < x < 4} 

(f) A = {a, l, m, n, p} 

    B = {q, r, l, a, s, n} 


6. Let X = {2, 4, 5, 6}   Y = {3, 4, 7, 8}   Z = {5, 6, 7, 8}, find

(a) (X - Y) ∪ (Y - X) 

(b) (X - Y) ∩ (Y - X) 

(c) (Y - Z) ∪ (Z - Y) 

(d) (Y - Z) ∩ (Z - Y) 

Practice Test on Operations on Sets

7. Let ξ = {1, 2, 3, 4, 5, 6, 7} and A = {1, 2, 3, 4, 5} B = {2, 5, 7} show that 

(a) (A ∪ B)' = A' ∩ B'

(b) (A ∩ B)' = A' ∪ B'

(c) (A ∩ B) = B ∩ A 

(d) (A ∪ B) = B ∪ A

8. Let P = {a, b, c, d}   Q = {b, d, f}   R = {a, c, e} verify that 

(a) (P ∪ Q) ∪ R = P ∪ (Q ∪ R) 

(b) (P ∩ Q) ∩ R = P ∩ (Q ∩ R) 


Answers for practice test on operations on sets are given below to check the correct answers. 

Answers: 

1. (a) {2, 3, 4, 5, 6, 7} 
(b) {2, 3, 4, 5, 6, 7, 8, 9} 
(c) {4, 5, 6, 7, 8, 9} 
(d) {4, 5, 6, 7, 8, 9, 10, 11}
(e) {2, 3, 4, 5, 6, 7, 8, 9} 
(f) {2, 3, 4, 5, 6, 7, 8, 9} 
(g) {4, 5, 6, 7, 8, 9, 10, 11} 

2. (a) {8, 10, 12}
(b) {12, 14}
(c) ∅
(d) {12} 
(e) d 
(f) {8, 10, 12, 14, 16} 
(g) {8} 
(h) {8, 10, 12, 14} 
(i) {8, 10, 12, 16} 


3. (a) {4, 10, 16, 19, 22} 
(b) {6, 11, 21} 
(c) {6, 11, 16, 21}
(d) {4, 7, 10, 13, 19, 22}
(e) {20} 
(f) {3, 5, 7, 9, 13, 15, 17} 
(g) {5, 19, 17, 20} 
(h) {5, 9, 13, 17, 20} 
(i) {6, 16, 21}
(j) {4, 7, 10, 16, 19, 22} 
(k) {3, 7, 11, 15} 
(l) {3, 5, 9 11, 15, 17}


4. B 

5. (a) {F, E, A, S, T}, {E, A, S, T}, {F} 
(b) {1, 2, 3, 4, 5, 6, 7, 8}, {5, 6, 7}, {1, 2, 3, 4} 
(c) {1, 2, 3, 4, 6, 8, 10, 12}, {2, 4, 6}, {1, 3, 12} 
(d) {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}, d, {2, 4, 6, 8, 10} 
(e) {-1, 0, 1, 2, 3}, {0, 1}, {-1}
(f) {a, 1, m, n, p, q, r, s}, {a, l, n}, {m, p} 


6. (a) {2, 3, 5, 6, 7, 8} 
(b) d 
(c) d {3, 4, 5, 6}
(d) d


7. (a) L.H.S. = R. H. S = {6} 
(b) L.H.S. = R. H. S = {1, 3, 4, 6, 7} 
(c) {2, 5} 
(d) {1, 2, 3, 4, 5, 7}


8. (a) {a, b, c, d, e, f} 
(b) d