Matematika terapan week 3. set

2012/2013 M. Ilyas Hadikusuma, M.Eng Matematika Terapan 1

TIF 21101

APPLIED MATH 1

(MATEMATIKA TERAPAN 1)

Week 3

SET THEORY

(Continued)

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORY

OBJECTIVES:

1. Subset and superset relation

2. Cardinality, Power of Set, Venn diagram

3. Algebra Law of Sets

4. Inclusion

5. Cartesian Product

Subset & superset relationWe use the symbols of:

⊆ � is a subset of

⊇ � is a superset of

We also use these symbols⊂ � is a proper subset of

⊃ � is a proper superset of

Why they are different?

They maen……

S⊆T means that every element of S is also an element of T.

S⊇T means T⊆S.

S⊂T means that S⊆T but .

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 ⊆ A A ⊄ B A ⊄ C

B ⊂ A B ⊆ B B ⊂ C

C ⊄ A C ⊄ B C ⊆ C

Prove them !!!

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,

|{{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}}.

Example:

A = {a, b, c} where |A| = 3

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??

SET THEORYSET THEORYSET THEORYSET THEORY

Venn DiagramWe use venn diagram to pictoral representation of

– studied and taught logic and probability theory

– articulated Boole’s algebra of logic

– devised a simple way to diagram set operations (Venn Diagrams)

John Venn 1834-1923

Universal Set

Sets A & B

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

Diketahui : S = { x | 10 < x ≤ 20, x ∈ B }

M = { x | x > 15, x ∈ S }

N = { x | x > 12, x ∈ S }Gambarlah diagram vennya

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:

SET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYSET THEORYAlgebra Law of Sets

Inclusion and Exclusion of Sets

For A and B, Let A and B be any finite sets. Then :

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.

A ∪ B = A + B – A ∩ B

Inclusion Exclusion

• How many elements are in A∪B?|A∪B| = |A| + |B| − |A∩B|

• Example:

{2,3,5}∪{3,5,7} = {2,3,5,3,5,7} ={2,3,5,7}

Exercise :

From 32 people who save paper or bottles (or both) for recycling, 30 save paper and 14 save bottles. Find the number of people who

(a) save both,

(b) save only paper, and

(c) save only bottles.

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|.

• Note that the Cartesian product is not

commutative: A×B ≠ B×A.

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