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HIMPUNAN & BILANGAN

Segaf, SE.MSc. 

Mathematical Economics

Economics Faculty

State Islamic University

Maulana Malik Ibrahim Malang

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PREFACE

Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is.

Rather it is an approach to economics analysis by using mathematical symbols in the statement of the problem and also draws upon known mathematical theorems to aid in reasoning.

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icsMATHEMATICAL VS. NONMATHEMATICAL ECONOMICS Advantage of Language of math in

economicsPrecise(accurate), concise (to the point)Draws on math theorems to show the wayForces declaration of assumptionsAllow treatment of the n-variable caseLanguage as a form of logic

Too much rigor(inflexibility) and too little reality

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BILANGAN NYATA (REAL NUMBERS)

Bilangan Nyata(Real #s)

Rational #s

Bilangan Bulat(Integer

s)

Negative integers

Zero

Positive integers

Bilangan

Pecahan

(Fractions)

Irrational #s

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THE REAL-NUMBER SYSTEM

Real numbers (universal set, continuous,+, -, 0) Irrational numbers

Those #s that can’t be expressed as a ratio, e.g., sqrt. 2, pi,

Rational numbersFractions: can be expressed as ratio of integers

Integers: expressed as whole numbers (or ratio of itself to 1)

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THE CONCEPT OF SETS (HIMPUNAN)

A Sets (Himpunan) is simply a collection of distinct objects.

Set Notation (Penulisan Himpunan) Enumeration (satu per satu)

Example: “S” represent of three numbers 3,8, and 9, we can write by enumeration:

S = {3, 8, 9} Description

Example: “I” denote of all positive integers, we may describe by write :

I = {x I x a positive integers}

Membership of a set denotes by symbol ∈ 3 ∈ S do ∉ Y 10 ∈ K

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HUBUNGAN DIANTARA HIMPUNAN

If, S1 = {2,7,a,f} and S2 = {2,a,f,7} S1 = S2

If, S = {6,5,10,4,2} and T = {10,5} T adalah himpunan bagian dari S (subset of S), jika dan hanya jika x ∈ T memenuhi x ∈ S, we may write: T ⊂ S (T is Contained in S) S ⊃ T (S includes T)

Can we say S1 ⊂ S2 and S2 ⊂ S1 ? Null set or empty set denotes by ∅ or { }. Is

it different with zero ? Himpunan kosong ∅ atau { } juga

merupakan himpunan bagian dari setiap himpunan apapun.

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OPERATIONS OF SETS

Add, subtract, multiply, divide, Square root of some numbers mathematical operation.

Three principal of mathematical operation for a set of numbers involved : the union (gabungan), intersection (irisan) and complement (pelengkap) of sets.

If, A = {3, 5, 7} and B = {2, 3, 4, 8} untuk mengambil gabungan dari dua himpunan A dan B (to take the union of two sets A and B) perlu dibentuk himpunan baru yang berisi elemen-elemen yang dimiliki A maupun B. Himpunan gabungan A dan B menggunakan simbol A ∪ B. Hence A ∪ B = {2, 3, 4, 5, 7, 8}

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CONT- OPERATION OF SETS

Irisan (intersection) himpunan A dan B adalah a new sets which contains those elements (and only those element) belonging to both A dan B.

The intersection sets A and B symbolized by A ∩ B, from the example above A ∩ B = {3}

When A = {-3, 6, 10} and B = {9, 2, 7, 4} A ∩ B = ∅ the set of A and B are disjoint.

formal definitions of “union and intersection” are: Intersection : A ∩ B = {x I x ∈ A and x ∈ B} Union : A ∪ B = {x I x ∈ A or x ∈ B}

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PICT-1 (VENN DIAGRAMS)

A B

A

B

A ∩ B

Irisan

A ∪ B

(Gabungan )

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CONT2- OPERATION OF SETS

U himpunan universal himpunan besar dan terdiri dari beberapa himpunan bagian (Larger of set, contains of some sets).

Let say A = {3, 6, 7}; and U = {1, 2, 3, 4, 5, 6, 7} complement of set A (Ã) as the set of all numbers in the Universal Set U, that are not in the set of A Ã = {x I x ∈ U and x ∉ A} = {1,2,4,5}

Thus, what is the Complement of U?

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PICT2 (VENN DIAGRAMS)

Complement Ã

Ã

A

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PICT-3 (VENN DIAGRAMS)

A ∩ B ∩ C

A ∪ B ∪ C

To take the union of three sets A, B and C, first we take the any of two sets, then “union” the resulting set with the third. A similar procedure is applicable to the intersections operation.

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THE LAW OF SETS OPERATION (DALIL-DALIL HIMPUNAN)

Dalil-Dalil Operasi

Himpunan

Hukum Kumutatif

(Cumutative law)

Hukum Asosiatif

(Associative law)

Hukum Distributif

(Distributive law)

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CONT. THE LAW OF SETS OPERATION (DALIL-DALIL HIMPUNAN)

See pict 1 (slide 10) at union diagram A ∪ B and B ∪ A At intersection diagram A ∩ B and B ∩ A Called : CUMUTATIVE LAW

See pict 3 (slide 13) A ∪ (B ∪ C) = (A ∪ B) ∪ C A ∩ (B ∩ C) = (A ∩ B) ∩ C Called : ASSOCIATIVE LAW

What about the combination of union and intersections? A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C) Called : DISTRIBUTIVE LAW

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EXERCISE

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