The document is a lecture on set theory that covers several topics: 1. It defines subset and superset relations, and introduces the symbols for subset, superset, proper subset, and proper superset. 2. It discusses cardinality and the power set of a set, defining the cardinality of a set as the number of elements it contains and the power set as the set of all subsets. 3. It explains the inclusion-exclusion principle for calculating the size of the union of two finite sets.

1. TIF 21101
APPLIED MATH 1
(MATEMATIKA TERAPAN 1)
Week 3
SET THEORY
(Continued)
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

2. SET THEORY
OBJECTIVES:
1. Subset and superset relation
2. Cardinality & Power of Set
3. Algebra Law of Sets
4. Inclusion
5. Cartesian Product
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

3. SET THEORY
Subset & superset relation
We use the symbols of:
Í is a subset of
Ê is a superset of
We also use these symbols
Ì is a proper subset of
É is a proper superset of
Why they are different?
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

4. SET THEORY
They maen……
SÍT means that every element of S is also
an element of T.
SÊT means TÍS.
SÌT means that SÍT but .
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

5. SET 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
• C = {2, 6, 8, 4}
• Subset Relationships
A Í A A Ë B A Ë C
B Ì A B Í B B Ì C
C Ë A C Ë B C Í C
Prove them !!!
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

6. SET 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., |Æ|=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}) = {Æ, {a}, {b}, {a,b}}.
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

7. SET THEORY
Example:
A = {a, b, c} where |A| = 3
P (A) = {{a, b}, {a, c}, {b, c}, {a}, {b}, {c}, A, f}
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??
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

8. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

9. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

10. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

11. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

12. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

13. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

14. SET 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
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
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

15. SET THEORY
Diketahui : S = { x | 10 x 20, x Î B }
M = { x | x 15, x Î S }
N = { x | x 12, x Î S }
Gambarlah diagram vennya
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

16. SET THEORY
Jawab : S = { x | 10 x 20, x Î B } = { 11,12,13,14,15,16,17,18,19,20 }
M = { x | x 15, x Î S } = { 16,17,18,19,20}
N = { x | x 12, x Î S } = { 13,14,15,16,17,18,19,20}
M Ç N = { 16,17,18,19,20 }
Diagram Vennya adalah sbb:
N M
16
17
18
19
20
13
14 15
S
11
12
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

17. SET THEORY
Algebra Law of Sets
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

18. SET THEORY
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

19. SET THEORY
Set’s Inclusion and Exclusion
For A and B, Let A and B be any finite sets. Then :
½A È B½ = ½A½ + ½B½ – ½A Ç B½
Inclusion Exclusion
In other words, to find the number n(A È B) of elements in the union
A È B, we add n(A) and n(B) and then we subtract n(A Ç B); that is,
we “include” n(A) and n(B), and we “exclude” n(A Ç B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A Ç B twice. This principle holds for any number of sets.
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

20. SET THEORY
Set’s Inclusion and Exclusion
For A and B, Let A and B be any finite sets. Then :
½A È B½ = ½A½ + ½B½ – ½A Ç B½
Inclusion Exclusion
In other words, to find the number n(A È B) of elements in the union
A È B, we add n(A) and n(B) and then we subtract n(A Ç B); that is,
we “include” n(A) and n(B), and we “exclude” n(A Ç B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A Ç B twice. This principle holds for any number of sets.
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

21. SET THEORY
Inclusion and Exclusion of Sets
For A and B, Let A and B be any finite sets. Then :
½A È B½ = ½A½ + ½B½ – ½A Ç B½
Inclusion Exclusion
In other words, to find the number n(A È B) of elements in the union
A È B, we add n(A) and n(B) and then we subtract n(A Ç B); that is,
we “include” n(A) and n(B), and we “exclude” n(A Ç B). This follows
from the fact that, when we add n(A) and n(B), we have counted the
elements of A Ç B twice. This principle holds for any number of sets.
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

22. Inclusion-Exclusion Principle
• How many elements are in AÈB?
|AÈB| = |A| + |B| − |AÇB|
• Example:
{2,3,5}È{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

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 ÇB)c = {tidak suka keduanya} n((A ÇB)c) = 5
Maka A ÇB = {suka keduanya}
n(A ÇB) = x
{siswa suka bakso saja} = 20 - x
{siswa suka siomay saja} = 46 - x
Perhatikan Diagram Venn berikut
S
A 20 - x x 46 - x B
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1
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 THEORY
Berapa banyaknya bilangan bulat antara 1
dan 100 yang habis dibagi 3 atau 5?
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

25. Cartesian Products of Sets
• For sets A, B, their Cartesian product
A×B :º {(a, b) | aÎA Ù bÎB }.
• E.g. {a,b}×{1,2} = {(a,1),(a,2),(b,1),(b,2)}
• Note that for finite A, B, |A×B|=|A||B|.
• Note that the Cartesian product is not
commutative: A×B B×A.
2014/2015 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1