Content:
Set operations
Venn diagram and application up to 3 set problem.
Operations on Sets
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 B. 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}
https://youtu.be/uT9WC6lVpK0
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
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.
https://youtu.be/xZELQc11ACY
EVALUATION
1. Given that ξ= {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
(vi) (P ⋃Q) ∩(Q ⋃R)
(vii) (P ∩Q) ⋃(Q ∩R)
2. If A = {1, 2, 3, 4} and
B = {3, 5, 6}, find:
(i) A ∩B (ii) A⋃ B
(iii) (A ∩ B)⋃ B
(iv) (A ⋃ B) ∩ A (WAEC)
3. 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}
(c) C = {a, u, m, n} (d) P = {u}
(e) Q = { } or Ø (f) ξ
4. If A = {7, 8, 9, 10}, B = {8, 10, 12, 14}
and C = {7, 9, 10, 14. 15} find the following:
(a) A ⋃B (b) B⋃ C
(c) A ⋃B ⋃ C (d) A ∩B
(e) A ∩C (f) B ∩C
(g) (A⋃ B) ∩(A⋃ C)
(h) A ∩(B⋃ C)
5. 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)
(b) A∩( B⋃ C)
(c) B ∩(A⋃ C)
(d) Show that:
(A∩B) ⋃ (A∩C) = A ∩(B⋃ C)
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 Ac or A′. for example
If ξ = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10} and A = {3, 5, 6, 9} then Ac or A′ = {1, 2, 4, 7, 8, 10}
Using Venn diagram, this is represented by the shaded portion below:
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′
(vii) (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:
(i) A′ (ii) B′
(iii) A′⋃B′ (iv) A′∩B′
(v) (A ∩B)′ (vi) (A ⋃B)′
(vii) (B′)′
Solution:
(i) A′ = {b, d, f, h, j}
(ii) B′ = {a, e, g, h}
(iii) A′⋃B′ = {a, b, d, e, f, g, h, j}
(iv) A′∩B′ = { h }
(v) A ∩B ={c, i }
(A ∩B)′ ={a, b, d, e, f, g, h, j}
(vi) A ⋃B = {a, b, c, d, e, f, g, i, j}
(A ⋃B)′= { h}
(vii) B′= {a, e, g, h}
(B′)′ = {b, c, d, f, i, j} = B
From the example above, we also observe that A′⋃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′
(ii) (A ∩B)′ = A′⋃B′
(iii) (A′)′ = A or (B′)′ = B
These are known as De Morgan’s Laws of Complementation.
https://youtu.be/fLMN0wtiz-4
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).
https://youtu.be/N_fZwQjsZvs
EVALUATION
1. If ξ = {x : 0 < x ≤ 15 x εƵ}
A = {1, 5, 7, 11} and
B = {5, 7, 14}
Find the following:
(a) A ∩B (b) A′ (c) B′
(d) (A ⋃B)′ (e) A′∩B (f)A ∩B′
(g) Show that: (A∩B)′ = A′⋃ B′
2. 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′
3. 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
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.
https://youtu.be/gomAB2FYreM
PROBLEMS INVOLVING TWO SETS.
For two intersecting sets, the diagram is given below with the labels of what each compartment represents.
Compartment I:represent the set of elements in A only. i.e. A B/ using set notations.
Compartment II: represents the set of elements common to both A and B i.e. AB
Compartment III: represents the set of elements in B only i.e. A/ B
Compartment IV: represents the set of elements that are neither in A nor B i.e.
(AB) /or A/ B/
https://youtu.be/oSLitQKUPiY
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() = 40
n(L) = 19
n(B) = 17
n(L B)/ = 13
Let x represents those that have visited both Cities i.e. n(LB) = x
L B
19 – x x 17 – x
13
19 – x + x +17 – x + 13 = 40
49 – x = 40
49 – 40 = x
9 = x
9 Students have visited both Cities
(b) 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
(i) those that offer neither.
(ii)the number that offers Biology but not
Agricultural Science (i.e. Biology only)
Solution:
(i) n() = 45
n(A) = 21
n(B) = 25
n(A B) = 6
Let n(AB) / = x
i.e. Let those that offer neither be x.
A B
21 – 6
6 25 – 6
x
21 - 6 + 6 +25 – 6 + x = 45
15 + 6 + 19 + x = 45
40 + x = 45
x = 45 – 40
x = 5
Those that offer neither,
i.e. n(AB) /= 5
https://youtu.be/0h8a3POgKxU
EVALUATION
(1). 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) Hausa.
(ii)Yoruba only
(iii)Hausa only.
(2) 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
(i) both fried rice and salad?
(ii) salad but not fried rice?
(ii) neither fried rice nor salad?
(3) The Venn diagram below represents a universal set of integers and its subsets
P and Q. List the elements of the following sets;
(a) P Q
(b) P Q
P 1 9
(c ) Q 2 3 8, 4
(d) P 10 7 6
5 Q
SSCE, NOV. 1995 Nọ 1. (WAEC)
PROBLEMS INVOLVING THREE SETS.
The Venn diagram is made up of eight compartments as shown below:
CompartmentI represents ABC
(elements common to the three sets A, B and C).
Compartment II represents ABC/
(elements common to both A and B only).
CompartmentIII represents AB/C
(elements common to both A and C only).
Compartment IV represents A/BC
(elements common to both B and C only).
CompartmentV represents AB/C/
(elements of A only).
CompartmentVI represents A/ B C /
(elements of B only).
Compartment VII represents A/B/C
(elements of C only).
Compartment VIII represents (ABC)/ or A/B/C/ 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.
(a) Represent the above information in a Venn diagram
(b) How many people play the three games?
(c) 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() = 80
n(VF/H/) i.e. Volleyball only = 15
n(V/FH/) i.e. Football only = 18
n(V/F/H) i.e. Handball only = 21
n(VFH/) i.e. Volleyball and Football only = 5
n(VF/H) i.e. Volleyball and Handball only = 8
n(V/FH) i.e. Football and Handball only =10
Let n(VFH) = 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: list out all information given in the question.
Let V be Volley ball
F be Football
H be Hand ball
(a) n() = 80
n(V) = 31
n(F) = 36
n(H) = 42
n(V F H) = 80 (Since each play at least one of the games).
n(V F) = 8
n(V H) = 11
n(F H) = 13.
Let n(V F 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 F H/)
(Since x is already in the circle of V F)
= n(V F) – x
= 8 – x
For Volleyball and Handball only
i.e. n(V F/ H)
(Since x is already in the circle of V H)
= n(V H) – x
= 11 – x
For Football and Handball only
i.e. n(V/ F H)
(Since x is already in the circle of F H)
= n(F H) – x
= 13 – x
For Handball only
i.e. n(V/ F/ 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
H
For Football only i.e. n(V/ F 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
8 - x F
x
13 – x
For Volleyball only i.e.n (V F/ H/)
n(V) – (All values already written in the circle of Volley ball).
= 31 – [ (8 – x ) + x + ( 11 – x ) ]
= 31 – [19 – x] V
= 31 – 19 + x 8 - x
= 12 + x x
11 – x
(b) To get the value of x, which represent those that play all three games, weadd 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.
i.e.
V F
15 5 18
3
8 10
21
H
b (ii) Exactly two of the three games
= n(V F H/) + n(V F/ H) + n(V/ F 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 F/ H/) + n(V/ F H/) + n(V/ F/ 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/F/ H) = 18 + x
= 18 + 3
= 21
21 of them play Handball only.
https://youtu.be/OG045QBuUrg
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.
(a) Draw a Venn diagram to illustrate the information above using x to represent those that took the three.
(b) Find the:
(i) Value of x
(ii) Number that took Botany
(iii) Number that took Zoology and Chemistry.
(2). 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:
(i) All the three games;
(ii) 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
(i) C A/
(ii) A/ (B C)
WASSCE, June 2002, No. 2
ASSIGNMENT
New General Mathematics for SSS, Book 1 Pages 101 – 104 Exercise 8d
Question no. 7,9,11,12,13, 14 and 15
READING
3. New General Mathematics for SSS, Book 1 Pages 97 – 104.
4. Man Mathematics for SSS, Book 1, Pages 39- 60
REFERENCES
1. M.F Macrae etal (2011),New General Mathematics for Senior Secondary Schools 1.
2. MAN Mathematics for senior Secondary Schools 1.
3. New school mathematics for senior secondary school et al; Africana publishers limited
4. Fundamental General Mathematics For Senior Secondary School by Idode G. O