Set operations Venn diagram and application up to 3 set

Subject :

Mathematics

Topic :

Logarithms

Class :

SS 1

Term:

First Term

Week:

Week 8

Instructional Materials:

Wall charts
Online Resources
Pictures

Related Audio Visual
Mathematics Textbooks

Reference Materials

Scheme of Work
Online Information

Textbooks
Workbooks
Education Curriculum

Previous Knowledge :

The pupils have previous knowledge of

Definition of sets , Set notations and Types of sets

Behavioural Objectives: At the end of the lesson, the students should be able to

define Venn diagram and application up to 3 set
Set operations

Content:

WEEK 8:

DATE……………………….

TOPIC: SETS

Content:

Set operations
Venn diagram and application up to 3 set

SUB-TOPIC 5: Operations on Sets

A Venn diagram is a graphical way of representing the relationships between different sets of data. It is made up of a series of overlapping circles, each of which represents a set of data. The degree to which the circles overlap indicates the relationship between the different sets of data.

For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, we can represent the relationship between these two sets of data using a Venn diagram. The overlap between the two sets of data would indicate that the numbers 2 to 5 are a subset of the numbers 1 to 10.

We can also use Venn diagrams to represent the relationships between more than two sets of data. For example, if we have a set of data that represents the numbers 1 to 10, and another set that represents the numbers 2 to 5, and another set that represents the numbers 6 to 8, we can represent the relationships between these three sets of data using a Venn diagram. The overlap between the three sets of data would indicate that the numbers 2 to 5 are a subset of the numbers 1 to 10, and that the numbers 6 to 8 are a subset of the numbers 1 to 10.

Venn diagrams can be used to represent relationships between any sets of data, no matter how large or small. They can be used to represent relationships between sets of numbers, sets of people, sets of objects, or any other type of data.

There are a few applications of Venn diagrams. One application is to use Venn diagrams to test whether two sets of data are equivalent. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, we can use a Venn diagram to test whether these two sets of data are equivalent. To do this, we would draw a Venn diagram with two overlapping circles, one for each set of data. If the overlap between the two sets of data is equal to the size of one of the sets of data, then we can say that the two sets of data are equivalent.

Another application of Venn diagrams is to use them to find the union of two sets of data. The union of two sets of data is the set of all elements that are in either set. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, the union of these two sets of data would be the set of all numbers from 1 to 10. We can find the union of two sets of data by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the union of the two sets of data.

Finally, Venn diagrams can also be used to find the intersection of two sets of data. The intersection of two sets of data is the set of all elements that are in both sets. For example, if we have a set of data that represents the numbers 1 to 10, and another set of data that represents the numbers 2 to 5, the intersection of these two sets of data would be the set of all numbers from 2 to 5. We can find the intersection of two sets of data by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the intersection of the two sets of data.

Venn diagrams can be a helpful tool for visualizing and understanding relationships between different sets of data. They can be used to test whether two sets of data are equivalent, to find the union of two sets of data, or to find the intersection of two sets of data. Venn diagrams can be used to visualize any type of relationship between any types of data.

1. What is a Venn diagram?

2. What are the applications of Venn diagrams?

3. How can Venn diagrams be used to test whether two sets of data are equivalent?

4. How can Venn diagrams be used to find the union of two sets of data?

5. How can Venn diagrams be used to find the intersection of two sets of data?

1. A Venn diagram is a graphical representation of the relationships between different sets of data.

2. Venn diagrams can be used to test whether two sets of data are equivalent, to find the union of two sets of data, or to find the intersection of two sets of data.

3. To test whether two sets of data are equivalent using a Venn diagram, the overlap between the two sets of data must be equal to the size of one of the sets of data.

4. The union of two sets of data is the set of all elements that are in either set. The union of two sets of data can be found by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the union of the two sets of data.

5. The intersection of two sets of data is the set of all elements that are in both sets. The intersection of two sets of data can be found by drawing a Venn diagram with two overlapping circles, one for each set of data. The area of the overlap between the two sets of data would be the intersection of the two sets of data.

The Union of Sets:

The Union of Sets A and B is the Set that is formed from the elements of the two Sets A and

This is usually denoted by “A ⋃B” meaning A Union B. Thus A ⋃B is the Set which consists of elements of A or of B or of both A and B.

Using Set notations, the Union of two Sets A and B is represented as follows

Example 10:

Given that A = {3, 7, 8, 10}

and B = {3, 5, 6, 8, 9} then

(A⋃ B) = {3, 5, 6,7, 8, 9,10}

Example 11:

If A = {a, b, c, d}, B = {1, 2, 3, 4} and C ={a, 3, θ} Then A ⋃B ⋃C = {a, b, c, d,1, 2, 3, 4, θ}

Intersection of Sets

The intersection of Sets A and B is the set of elements that are common to both A and B. This is usually denoted by “A ∩ B” meaning A intersection B. When represented using Venn diagram we have

If A ∩ B = Ø, then the Sets A and B are said to be disjoint. Disjoint Sets are Sets that have no element in common.

Example 12:

Given that A = {5, 7, 8, 10} and B = {3, 5, 6, 8, 9}, then A ∩ B = {5, 8}.

Example 13:

If P = {a, b, c, d, e, f, g}, Q = {b, c, e, g} and R = {a, c, d, f, g} Then, P ∩ Q ∩ R = {c, g}

Example 14:

If A = {1, 2, 3} and B = {6, 8, 10}, then

A ∩ B = { } or Ø. The Set A and B are disjoint.

EVALUATION

Given thatξ={21,22,23,24,…,29,30},

P = {21, 23, 25, 26, 28},

Q = {22, 24, 26, 27, 28} and

R = {21, 25, 26, 27, 30} are Subsets of ξ.. Find:

(i) P ⋃Q (ii) P ∩Q (iii) Q ∩R

(iv) (P ∩Q) ⋃R (v) P ∩Q ∩R

(P ⋃Q) ∩(Q ⋃R)

(P ∩Q) ⋃(Q ∩R)

If A = {1, 2, 3, 4} and

B = {3, 5, 6}, find:

A ∩B (ii) A⋃ B

(A ∩ B)⋃ B

(A ⋃ B) ∩ A (WAEC)

If ξ = {a, e, i, o, u, m, n} find the complement of the following Sets

(a) A = {e, o, u} (b) B= {i, o, u}

Q = { } or Ø (f) ξ

If A = {7, 8, 9, 10}, B= {8, 10, 12, 14}

and C= {7, 9, 10, 14. 15} find the following:

A ⋃B (b) B⋃ C

(e) A ∩C (f) B ∩C

(g) (A⋃ B) ∩(A⋃ C)

(h) A ∩(B⋃ C)

Let ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10},

A = {2, 3, 4, 5, 6,}

B = {3, 4, 6, 8, 10} and

C = {2, 4, 6, 7, 10}

Find:

(a) (A⋃ B) ∩(B⋃ C)

A∩( B⋃ C)
B ∩(A⋃ C)
Show that:

(A∩B) ⋃ (A∩C) = A ∩(B⋃ C)

SUB-TOPIC 6: The Complement of Set

If A is a Subset of the Universal Set ξ, then, the complement of the Set A are made up of elements that are not in A, but are found in the Universal Set ξ. This is usually denoted by A^c or A′. for example

If ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {3, 5, 6, 9} then A^c or A′ = {1, 2, 4, 7, 8, 10}

Using Venn diagram, this is represented by the shaded portion below:

Note that to find the complement of a Set, the Universal Set must be properly defined.

Example 15:

Given that

ξ = {11, 12, 13, 14, 15, 16, 17, 18, 19, 20} A = {11, 13, 16, 18, 20} and

B = {12, 14, 16, 18, 19, 20}.

Find the following:

(i) A′ (ii) B′ (iii) (A ⋃B)′

(iv) (A ∩B)′ (v) A′∩B′ (vi) A′⋃B′

(A′)′

Solution:

(i) A′ = {12, 14, 15, 17, 19}

(ii) B′ = {11, 13, 15, 17}

(iii) A ⋃B = {11, 12, 13, 14, 16, 18, 19, 20}

(A ⋃B)′= {15, 17}

(iv) A ∩B = {16, 18, 20}

(A ∩B)′ = {11, 12, 13, 14, 15, 17, 19}

(v) (A′∩B′) = {15, 17}

(vi) A′⋃ B′ = {11, 12, 13, 14, 15, 17, 19}

(vii) A′ = {12, 14, 15, 17, 19}

(A′)′ = {11, 13, 16, 18, 20} = A

NB:

From the example above, observe that from (iii) and (v) (A ⋃B)′ = A′∩B′ Also, from (iv) and (vi)

(A ∩B)′ = A′⋃B′ and from (vii)(A′)′ = A

Example 16:

Given that ξ = {a, b, c, d, e, f, g, h, i, j} A = {a, c, e, g, i} and

B = {b, c, d, f, i, j}.

Find the following:

A′ (ii) B′

(iii) A′⋃B′ (iv) A′∩B′

(v) (A ∩B)′ (vi) (A ⋃B)′

(vii) (B′)′

Solution:

A′ = {b, d, f, h, j}
B′ = {a, e, g, h}
A′⋃B′ = {a, b, d, e, f, g, h, j}
A′∩B′ = { h }
A ∩B ={c, i }

(A ∩B)′ ={a, b, d, e, f, g, h, j}

A ⋃B = {a, b, c, d, e, f, g, i, j}

(A ⋃B)′= { h}

B′= {a, e, g, h}

(B′)′ = {b, c, d, f, i, j} = B

From the example above, we also observe thatA′⋃B′ = (A ∩B)′ — From (iii) and (v)

A′∩B′ = (A ⋃B)′— From (iv) and (vi)

And (B′)′ = B — From (vii)From the last two examples we can clearly see that (A⋃B)′ = A′∩B′,

(A ∩B)′ = A′⋃B′ and (A′)′ = A

Generally, for any two Subsets A and B of a Universal Set ξ, the following are true:

(i) (A ⋃B)′ = A′∩B′

(A ∩B)′ = A′⋃B′
(A′)′ = A or (B′)′ = B

These are known as De Morgan’s Laws of Complementation.

Equality of Set

Two Sets A and B are said to be equal if they have exactly the same elements. This means that every element of the first Set also belongs to the second Set and vice versa. E.g.

If A = {2, 3, 4, 5} and B = {2, 3, 4, 5} then the Set A = the Set B because n(A) = n (B) = 4 and all members of the Set A are also members of the Set B.

If P = {8, 2, 4} and Q = {2, 4, 8} the Set P = Q (order of arrangement does not matter, the elements in both Sets are the same).

If A = {1, 2, 3} and B = {3, 5, 6} then A ≠ B.

Equivalent Sets

Two Sets are said to be equivalent if the Sets have equal number of elements. E.g.

If A = {2, 3, 4, 5} and B = {a, b, c, d} then the Sets A ≡ B (A is equivalents to B) since n(A) = n(B).

EVALUATION

If ξ={x : 0 < x ≤ 15 xεƵ}

A = {1, 5, 7, 11} and

B = {5, 7, 14}

Find the following:

A ∩B (b) A′ (c) B′

(d) (A ⋃B)′ (e) A′∩B (f)A ∩B′

(g) Show that: (A∩B)′ = A′⋃ B′

Given that ξ={x : 0 ≤ x < 10, x εƵ}

P = {1, 2, 4, 5}

Q = {2, 4, 6, 8} and

R = {1, 3, 4, 8, 9}

Find:

(a) P′ (b) Q′ (c) R′ (d) Q∩ R′

(e) P′∩ Q′ (f) P′⋃Q′ (g)(Q∩ R′)′

(h) (P′∩ R′)′ (i) (P ∩ ξ) ′

(j) P′∩ Q′∩ R′

P, Q and R are Subsets of the Universal Set µ such that µ = {3, 4, 5, 6, . . . , 16, 17, 18} P = {x : x is a number divisible by 3}

Q= {x : x is odd}

R = {x : x is a factor of 35}

Find: (i) P∩Q (ii) P⋃ Q′∩ R

(iii) Q⋃ R (iv) P′ (v) P′∩ R

SUB-TOPIC 7: THE USE OF VENN DIAGRAMS IN PROBLEM SOLVING

A Mathematician by name John Venn was the man to first represent the relationship between sets with diagrams. Ever since sets may be represented by diagrams called Venn diagrams.

The rectangle is used to represent the Universal set, and Circles for other sets, as we

shall see later.

PROBLEMS INVOLVING TWO SETS.

For two intersecting sets, the diagram is given below with the labels of what each compartment represents.

A

Compartment I:represent the set of elements in A only. i.e. A Ç B/ using set

notations.

Compartment II: represents the set of elements common to both A and B i.e. AÇB

Compartment III: represents the set of elements in B only i.e. A^/Ç B

Compartment IV: represents the set of elements that are neither in A nor B i.e. (AÈB) ^/or A^/Ç B^/

Example 15:

In a Survey of 40 Students in a class, 19 have visited Lagos and 17 have visited Benin City.

If 13 have visited neither. How many Students have visited:

(i) Both Cities; (ii) Benin City but not Lagos (i.e. Benin City only)

Solution:

n(x) = 40

n(L) = 19

n(B) = 17 n(L È B)^/ = 13

Let x represents those that have visited both Cities i.e. n(LÇB) = x

L B

19 – x x 17 – x

19 – x + x +17 – x + 13 = 40

49 – x = 40

49 – 40 = x

9 = x

\9 Students have visited both Cities

Those that have visited Benin City only are = 17 – x

= 17 – 9

= 8

\8 have visited Benin City only.

Example 16:

In a Class of 45 Students, if 21 offer Agricultural Science, 25 offer Biology and 6 offer both subjects. Find

those that offer
the number that offers Biology but not Agricultural Science (i.e. Biology only)

Solution:

(i) n(x) = 45 n(A) = 21 n(B) = 25

n(A ÇB) = 6

Let n(AÈB) ^/ = x

i.e. Let those that offer neither be x.

21 – 6 + 6 +25 – 6 + x = 45

15 + 6 + 19 + x = 45

40 + x = 45

\ x = 5

x = 45 – 40

\Those that offer neither,

i.e. n(AÈB) ^/= 5

EVALUATION

In a gathering of 30 people, x speak Hausa and 15 speak Yoruba. If 5 people speak both languages, find how many people that speak (i)

(ii)Yoruba only (iii)Hausa only.

In a Birthday party attended by 22

people, 10 ate fried rice and 13 ate salad. If x ate both fried rice and salad and (2x–5) ate none of the two. How many ate

both fried rice and salad?
salad but not fried rice?

(ii) neither fried rice nor salad?

The Venn diagram below represents a universal set xof integers and its subsets Pand Q. List the elements of the following sets;

P È Q
P Ç Q x

SSCE, NOV. 1995 Nọ 1. (WAEC)

SUB-TOPIC 8: PROBLEMS INVOLVING THREE SETS.

The Venn diagram is made up of eight compartments as shown below: x

C VIII

CompartmentI represents AÇBÇC (elements commontothethreesetsA,BandC). Compartment II represents AÇBÇC^/ (elements common to both A and B only).

CompartmentIII represents AÇB^/ÇC

(elements common to both A and C only).

Compartment IV represents A/ÇBÇC

(elements common to both B and C only).

CompartmentV represents AÇB^/ÇC^/ (elements of A only).

CompartmentVI represents A^/ÇBÇC^/ (elements of B only).

Compartment VII represents A^/ÇB^/ÇC (elements of C only).

Compartment VIII represents (AÈBÈC)^/ or A^/ÇB^/ÇC^/ elements that are not in any of the three sets but are in the Universal set.

Example 18:

The are 80 people in a sports camp. Each play at least one of the following games, volleyball, football and handball. 15 play volleyball only, 18 play football only, and 21 play handball only .If 5 play volleyball and foot ball only, 8 play volleyball and handball only, and 10 play football and Handball only.

Represent the above information in a Venn diagram
How many people play the three games?
How many people play football ?

Solution:

List of information given in the question is as follows

Let V be Volleyball

F be Football

H be Handball

n(x) = 80

n(VÇF/ÇH/) i.e. Volleyball only = 15 n(V/ÇFÇH/) i.e. Football only = 18 n(V/ÇF/ÇH) i.e. Handball only = 21 n(VÇFÇH/) i.e. Volleyball and Football only = 5 n(VÇF/ÇH) i.e. Volleyball and Handball only = 8 n(V/ÇFÇH) i.e. Football and Handball only =10

Let n(VÇFÇH) = x . i.e. Those that play the three games = x

(a)

(b) 15 + 5 + 18 + 8 + x + 10 + 21 = 80

77 + x = 80

x = 80 – 77

\ x = 3

\ 3 people play the three games.

(C) The number that plays football

n(F) = 18 + 5 + 10 + x

= 33 + 3

= 36

N.B

Since the word ONLY was used all through, the values are written directly in to each compartment without any manipulation as shown in the figure above.

Suppose the question 18 above is framed as shown in the example 19 below, then the Approach would be different.

Example 19:

There are 80 people in a sports camp and each plays at least one of the following games: volleyball, football and handball. 31 play volleyball, 36 play football and 42 play handball. If 8 play volleyball and football, 11 play volleyball and handball and 13 play football and handball. (a) Draw a Venn diagram to illustrate this information, Using x to represent the number that play the three games.

(b) How many of them play:

(i) All the three games,

(ii) Exactly two of the three games,

(iii) Exactly one of the three games (iv) handball only?

Solution:

Step 1: listoutallinformation giveninthequestion.

Let V be Volley ball

F be Football

H be Hand ball (a) n(x) = 80

n(V) = 31

n(F) = 36

n(H) = 42

n(V È F È H) = 80 (Since each play at least one of the games).

n(V Ç F) = 8

n(V Ç H) = 11

n(F Ç H) = 13.

Let n(V Ç F Ç H) = x.

To fill the Venn diagram we start with the centre Compartment

where n(V Ç F Ç H) = x.

How we obtained the value for each of the other compartments is shown below.

For Volleyball and football only

i.e. n(V Ç F Ç H/)

(Since x is already in the circle of V ∩ F)

= n(V Ç F) – x

= 8 – x

For Volleyball and Handball only

i.e. n(V Ç F/Ç H)

(Since x is already in the circle of V ∩ H)

= n(V Ç H) – x

= 11 – x

For Football and Handball only

i.e. n(V/Ç F Ç H)

(Since x is already in the circle of F ∩ H)

= n(F Ç H) – x

= 13 – x

For Handball only

i.e. n(V/Ç F/Ç H)

n(H) – (All values already written in the circle of Handball)

= 42 – [ (11 – x ) + x + (13 – x ) ]

= 42 – [24 – x ]

= 42 – 24 + x

= 18 + x x

11– x 13 – x

For Football only i.e. n(V/Ç F Ç H/)

n(F) – ( All values already written in the circle of Football)

= 36 – [ (8 – x) + x + (13 – x) ]

= 36 – [21 – x ]

= 36 – 21 + x

= 15 + x

For Volleyball only i.e.n (V Ç F/Ç H/)

n(V) – (All values already written in the circle of Volley ball).

= 31 – [ (8 – x ) + x + ( 11 – x ) ]

= 31 – [19 – x] V

(b) To get the value of x, which represent those that play all three games, we

add all the Compartments of the Venn diagram together and equate it to the total value in the Universal set and solve for x.

i.e. 12 + x + 8 – x + 15 + x + x + 11 – x + 13 – x + 18 + x = 80

77 + x = 80

x = 80 – 77

x = 3

\ 3 people play all three games

NOTE THAT

If this value, x =3, is substituted into the Venn diagram, the answer obtained in the previous example would be got.

b (ii) Exactly two of the three games

= n(V Ç F Ç H/) + n(V Ç F/Ç H) + n(V/Ç F Ç H)

= 8 – x + 11 – x + 13 – x

= 32 – 3x

= 32 – 3(3)

= 32 – 9

= 23

\ 23 of them play exactly two of the three games.

b (iii) Exactly one of the three games

= n(V Ç F/Ç H/) + n(V/Ç F Ç H/) + n(V/Ç F/Ç H)

= 12 + x + 15 + x + 18 + x

= 45 + 3x

= 45 + 3(3)

= 45 + 9

= 54

\ 54 of them play exactly one of the three games.

b (iv) For Handball only

n(V/ÇF/Ç H) = 18 + x

= 18 + 3

= 21

\21 of them play Handball only.

EVALUATION

(1) In a Class of 80 undergraduate Students, 21 took elective Courses from Botany only, 16 took from Zoology only, 13 took from Chemistry only. If each of the Students took elective from at least one of the above-mentioned Courses, 7 took Botany and Zoology only, 3 took Zoology and Chemistry only and 8 took Botany and Chemistry only.

Draw a Venn diagram to illustrate the information above using x to represent those that took the
Find the:

(i) Value of x

(ii) Number that took Botany

(iii) Number that took Zoology and Chemistry.

In a Group of 120 Students, 72 of them play Football, 65 play Table Tennis and 53 Play Hockey. If 35 of the Students play both Football and Table Tennis, 30 play both Football and Hockey, 21 play both Table Tennis and Hockey and each of the Students play at least one of the three games,

(a) Draw a Venn diagram to illustrate this information. (b) How many of them play:

All the three games;
Exactly two of the three games; (iii) Exactly one of the three games

(iv) Football alone?

SSCE, NOV. 1996, № 6 (WAEC).

(3) The set A = {1,3,5,7,9,11}, B = {2,3,5,7,11,15} and C={3,6,9,12,15} are subsets of

ε = {1,2,3,…,14,15} (a) draw a Venn diagram to illustrate the given information. (b) use your Venn diagram to find

C Ç A/
A/Ç (B È C)

WASSCE, June 2002, No. 2

WEEKEND ASSIGNMENT

New General Mathematics for SSS, Book 1 Pages 101 – 104 Exercise 8d

Question no. 7,9,11,12,13, 14 and 15

WEEKEND READING

New General Mathematics for SSS, Book 1 Pages 97 – 104. 4. Man Mathematics for SSS, Book 1, Pages 39- 60

REFERENCES

F Macraeetal (2011),New General Mathematics for Senior Secondary Schools 1.

6. MAN Mathematics for senior Secondary Schools 1.

New school mathematics for senior secondary school et al; Africana publishers limited

8. Fundamental General Mathematics For Senior Secondary School by Idode G. O

WEEK 11 Revision.

WEEK 12 Examinatio

Presentation

The topic is presented step by step

Step 1:

The subject teacher revises the previous topics

Step 2.

He or she introduces the new topic.

Step 3:

The subject teacher allows the pupils to give their own examples and he corrects them when the need arises.

Conclusion:

The subject teacher wraps up or concludes the lesson by giving out a short note to summarize the topic that he or she has just taught.

The subject teacher also goes round to make sure that the notes are well copied or well written by the pupils.

He or she makes the necessary corrections when and where the needs arise.

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Tags:First Term, MATHEMATICS

Mathematics

Logarithms

Definition of sets , Set notations and Types of sets

Content:

TOPIC: SETS

Example 10:

Example 11:

Example 12:

Example 13:

Example 14:

Example 15:

Solution:

NB:

Example 16:

Solution:

EVALUATION

SUB-TOPIC 7: THE USE OF VENN DIAGRAMS IN PROBLEM SOLVING

PROBLEMS INVOLVING TWO SETS.

A

Example 15:

Solution:

Example 16:

Solution:

EVALUATION

SSCE, NOV. 1995 Nọ 1. (WAEC)

Example 18:

Solution:

(C) The number that plays football

N.B

Example 19:

Solution:

Step 1: listoutallinformation giveninthequestion.

For Volleyball and football only

For Volleyball and Handball only

For Football and Handball only

For Handball only

For Football only i.e. n(V/Ç F Ç H/)

For Volleyball only i.e.n (V Ç F/Ç H/)

NOTE THAT

b (ii) Exactly two of the three games

EVALUATION

SSCE, NOV. 1996, № 6 (WAEC).

WASSCE, June 2002, No. 2

WEEKEND READING

REFERENCES

6. MAN Mathematics for senior Secondary Schools 1.

8. Fundamental General Mathematics For Senior Secondary School by Idode G. O

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