H IMPUNAN & B ILANGAN Segaf, SE.MSc. Mathematical Economics Economics Faculty State Islamic University Maulana Malik Ibrahim Malang 1 Mathematical Economics

P REFACE 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. 2 Mathematical Economics

M ATHEMATICAL VS. N ONMATHEMATICAL E CONOMICS 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 3 Mathematical Economics

B ILANGAN N YATA (R EAL N UMBERS ) Bilangan Nyata (Real #s) Rational #s Bilangan Bulat (Integers) Negative integers ZeroPositive integers Bilangan Pecahan (Fractions) Irrational #s 4 Mathematical Economics

T HE 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) 5 Mathematical Economics

T HE C ONCEPT OF S ETS (H IMPUNAN ) 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 6 Mathematical Economics

H UBUNGAN DIANTARA H IMPUNAN If, S 1 = {2,7,a,f} and S 2 = {2,a,f,7}  S 1 = S 2 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 S 1 ⊂ S 2 and S 2 ⊂ S 1 ? Null set or empty set denotes by ∅ or { }. Is it different with zero ? Himpunan kosong ∅ atau { } juga merupakan himpunan bagian dari setiap himpunan apapun. 7 Mathematical Economics

O PERATIONS OF S ETS 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} 8 Mathematical Economics

C ONT - O PERATION OF S ETS 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} 9 Mathematical Economics

P ICT -1 (V ENN D IAGRAMS ) AB A ∩ B Irisan A ∪ B (Gabungan ) 10 Mathematical Economics

C ONT 2- O PERATION OF S ETS 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? 11 Mathematical Economics

P ICT 2 (V ENN D IAGRAMS ) Complement à à A 12 Mathematical Economics

P ICT -3 (V ENN D IAGRAMS ) A ∩ B ∩ C A ∪ B ∪ C 13 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

T HE L AW OF S ETS OPERATION (D ALIL - D ALIL H IMPUNAN ) Dalil-Dalil Operasi Himpunan Hukum Kumutatif (Cumutative law) Hukum Asosiatif (Associative law) Hukum Distributif (Distributive law) 14 Mathematical Economics

C ONT. T HE L AW OF S ETS OPERATION (D ALIL -D ALIL H IMPUNAN ) 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 15 Mathematical Economics

EXERCISE 16 Mathematical Economics

17 Mathematical Economics