This document discusses key concepts of set theory including: 1. Subsets, supersets, proper subsets and proper supersets are defined using the symbols ⊆, ⊇, ⊂ and ⊃. 2. The cardinality of a set is a measure of its number of elements, and the power set of a set S contains all possible subsets of S. 3. The inclusion-exclusion principle states that for sets A and B, the number of elements in their union is the sum of their individual elements minus the number shared between them.

1. TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Week 3
SET THEORY
(Continued)

2. SET THEORYSET THEORY
OBJECTIVES:
1. Subset and superset relation
2. Cardinality & Power of Set
3. Algebra Law of Sets
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
3. Algebra Law of Sets
4. Inclusion
5. Cartesian Product

3. SET THEORYSET THEORY
Subset & superset relation
We use the symbols of:
⊆ is a subset of
⊇ is a superset of
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
We also use these symbols
⊂ is a proper subset of
⊃ is a proper superset of
Why they are different?

4. SET THEORYSET THEORY
They maen……
S⊆T means that every element of S is also
an element of T.
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
an element of T.
S⊇T means T⊆S.
S⊂T means that S⊆T but .

5. SET THEORYSET THEORY
Examples:
• A = {x | x is a positive integer ≤ 8}
set A contains: 1, 2, 3, 4, 5, 6, 7, 8
• B = {x | x is a positive even integer < 10}
set B contains: 2, 4, 6, 8
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
set B contains: 2, 4, 6, 8
• C = {2, 6, 8, 4}
• Subset Relationships
A ⊆ A A ⊄ B A ⊄ C
B ⊂ A B ⊆ B B ⊂ C
C ⊄ A C ⊄ B C ⊆ C
Prove them !!!

6. SET THEORYSET THEORY
Cardinality and The Power of Sets
|S|, (read “the cardinality of S”), is a measure of
how many different elements S has.
E.g., |∅|=0, |{1,2,3}| = 3, |{a,b}| = 2,
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
E.g., |∅|=0, |{1,2,3}| = 3, |{a,b}| = 2,
|{{1,2,3},{4,5}}| = ……
P(S); (read “the power set of a set S”) , is the set
of all subsets that can be created from given set S.
E.g. P({a,b}) = {∅, {a}, {b}, {a,b}}.

7. SET THEORYSET THEORY
Example:
A = {a, b, c} where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, φ}
and |P (A)| = 8
In general, if |A| = n, then |P (A) | = 2n
How about if the set of S is not finite ? So we say S infinite.
Ex. B = {x | x is a point on a line}, can you difine them??

8. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

9. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

10. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

11. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

12. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

13. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

14. SET THEORYSET THEORY
Langkah-langkah menggambar diagram venn
1. Daftarlah setiap anggota dari masing-masing himpunan
2. Tentukan mana anggota himpunan yang dimiliki secara bersama-sama
3. Letakkan anggota himpunan yang dimiliki bersama ditengah-tengah
4. Buatlah lingkaran sebanyak himpunan yang ada yang melingkupi
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
4. Buatlah lingkaran sebanyak himpunan yang ada yang melingkupi
anggota bersama tadi
5. Lingkaran yang dibuat tadi ditandai dengan nama-nama himpunan
6. Selanjutnya lengkapilah anggota himpunan yang tertulis didalam
lingkaran sesuai dengan daftar anggota himpunan itu
7. Buatlah segiempat yang memuat lingkaran-lingkaran itu, dimana
segiempat ini menyatakan himpunan semestanya dan lengkapilah
anggotanya apabila belum lengkap

15. SET THEORYSET THEORY
Diketahui : S = { x | 10 < x ≤ 20, x ∈ B }
M = { x | x > 15, x ∈ S }
N = { x | x > 12, x ∈ S }
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
N = { x | x > 12, x ∈ S }
Gambarlah diagram vennya

16. SET THEORYSET THEORY
Jawab : S = { x | 10 < x ≤ 20, x ∈ B } = { 11,12,13,14,15,16,17,18,19,20 }
M = { x | x > 15, x ∈ S } = { 16,17,18,19,20}
N = { x | x > 12, x ∈ S } = { 13,14,15,16,17,18,19,20}
M ∩∩∩∩ N = { 16,17,18,19,20 }
Diagram Vennya adalah sbb:
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
16
17
18
19
20
MN
13
14 15
S
11
12
Diagram Vennya adalah sbb:

17. SET THEORYSET THEORY
Algebra Law of Sets
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

18. SET THEORYSET THEORY
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

19. SET THEORYSET THEORY
Set’s Inclusion and Exclusion
For A and B, Let A and B be any finite sets. Then :
A ∪ B = A + B – A ∩ B
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
In other words, to find the number n(A ∪ B) of elements in the union
A ∪ B, we add n(A) and n(B) and then we subtract n(A ∩ B); that is,
we “include” n(A) and n(B), and we “exclude” n(A ∩ B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A ∩ B twice. This principle holds for any number of sets.
Inclusion Exclusion

20. SET THEORYSET THEORY
Set’s Inclusion and Exclusion
For A and B, Let A and B be any finite sets. Then :
A ∪ B = A + B – A ∩ B
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
In other words, to find the number n(A ∪ B) of elements in the union
A ∪ B, we add n(A) and n(B) and then we subtract n(A ∩ B); that is,
we “include” n(A) and n(B), and we “exclude” n(A ∩ B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A ∩ B twice. This principle holds for any number of sets.
Inclusion Exclusion

21. SET THEORYSET THEORY
Inclusion and Exclusion of Sets
For A and B, Let A and B be any finite sets. Then :
A ∪ B = A + B – A ∩ B
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
In other words, to find the number n(A ∪ B) of elements in the union
A ∪ B, we add n(A) and n(B) and then we subtract n(A ∩ B); that is,
we “include” n(A) and n(B), and we “exclude” n(A ∩ B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A ∩ B twice. This principle holds for any number of sets.
Inclusion Exclusion

22. Inclusion-Exclusion Principle
• How many elements are in A∪B?
|A∪B| = |A| + |B| − |A∩B|
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
• Example:
{2,3,5}∪{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}

23. Contoh:
Dari 60 siswa terdapat 20 orang suka bakso, 46 orang suka siomay dan 5
orang tidak suka keduanya.
a. Ada berapa orang siswa yang suka bakso dan siomay?
b. Ada berapa orang siswa yang hanya suka bakso?
c. Ada berapa orang siswa yang hanya suka siomay?
Jawab: N(S) = 60
Misalnya : A = {siswa suka bakso} n(A) = 20
B = {siswa suka siomay} n(B) = 46
(A ∩∩∩∩B)c = {tidak suka keduanya} n((A ∩∩∩∩B)c) = 5
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
Maka A ∩∩∩∩B = {suka keduanya}
(A ∩∩∩∩B)c = {tidak suka keduanya} n((A ∩∩∩∩B)c) = 5
n(A ∩∩∩∩B) = x
{siswa suka bakso saja} = 20 - x
{siswa suka siomay saja} = 46 - x
Perhatikan Diagram Venn berikut
xA B20 - x 46 - x
S
5
n(S) = (20 – x)+x+(46-x)+5
60 = 71 - x
X = 71 – 60 = 11
a. Yang suka keduanya adalah x
= 11 orang
b. Yang suka bakso saja adalah
20-x = 20-11= 9 orang
c. Yang suka siomay saja adalah
46-x = 46-11= 35 orang

24. SET THEORYSET THEORY
Berapa banyaknya bilangan bulat antara 1
dan 100 yang habis dibagi 3 atau 5?
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng

25. Cartesian Products of Sets
• For sets A, B, their Cartesian product
A×B :≡ {(a, b) | a∈A ∧ b∈B }.
• E.g. {a,b}×{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• Note that for finite A, B, |A×B|=|A||B|.
Matematika Terapan 12014/2015 M. Ilyas Hadikusuma, M.Eng
• Note that for finite A, B, |A×B|=|A||B|.
• Note that the Cartesian product is not
commutative: A×B ≠ B×A.