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Himpunan & Bilangan

Himpunan & Bilangan. Segaf , SE.MSc . Mathematical Economics Economics Faculty State Islamic University Maulana Malik Ibrahim Malang. Preface. Mathematical economics is not a distinct branch of economics in the sense that public finance or international trade is.

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Himpunan & Bilangan

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  1. Himpunan& Bilangan Mathematical Economics Segaf, SE.MSc.Mathematical Economics Economics Faculty State Islamic University MaulanaMalik Ibrahim Malang

  2. 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. Mathematical Economics

  3. Mathematical vs. Nonmathematical Economics • Advantage of Language of math in economics • Precise(accurate), concise (to the point) • Draws on math theorems to show the way • Forces declaration of assumptions • Allow treatment of the n-variable case • Language as a form of logic • Too much rigor(inflexibility) and too little reality Mathematical Economics

  4. BilanganNyata (Real Numbers) Mathematical Economics

  5. 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 numbers • Fractions: can be expressed as ratio of integers • Integers: expressed as whole numbers (or ratio of itself to 1) Mathematical Economics

  6. The Concept of Sets (Himpunan) • A Sets (Himpunan) is simply a collection of distinct objects. • Set Notation (PenulisanHimpunan) • 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 Mathematical Economics

  7. HubungandiantaraHimpunan • 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 adalahhimpunanbagiandari S (subset of S), jikadanhanyajika 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 ? • Himpunankosong∅ atau { } jugamerupakanhimpunanbagiandarisetiaphimpunanapapun. Mathematical Economics

  8. 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}  untukmengambilgabungandariduahimpunan A dan B (to take the union of two sets A and B) perludibentukhimpunanbaru yang berisielemen-elemen yang dimiliki A maupun B. Himpunangabungan A dan B menggunakansimbol A ∪ B. Hence A ∪ B = {2, 3, 4, 5, 7, 8} Mathematical Economics

  9. Cont- Operation of Sets • Irisan (intersection)himpunan A dan B adalaha 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} Mathematical Economics

  10. Pict-1 (Venn Diagrams) A ∩ B Irisan A ∪ B (Gabungan ) Mathematical Economics

  11. Cont2- Operation of Sets • U  himpunan universal  himpunanbesardanterdiridaribeberapahimpunanbagian (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? Mathematical Economics

  12. Pict2 (Venn Diagrams) Complement à Mathematical Economics

  13. 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. Mathematical Economics

  14. The Law of Sets operation (Dalil-DalilHimpunan) Mathematical Economics

  15. Cont. The Law of Sets operation (Dalil-DalilHimpunan) • 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 Mathematical Economics

  16. exercise Mathematical Economics

  17. Mathematical Economics

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