TIF APPLIED MATH 1 (MATEMATIKA TERAPAN 1) Week 3 SET THEORY (Continued)

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1 TIF APPLIED MATH 1 (MATEMATIKA TERAPAN 1) Week 3 SET THEORY (Continued)

2 OBJECTIVES: 1. Subset and superset relation 2. Cardinality & Power of Set 3. Algebra Law of Sets 4. Inclusion 5. Cartesian Product

3 Subset & superset relation We 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?

4 They maen SET THEORY 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.

5 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!!!

6 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}}.

7 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) = 2 n 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??

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14 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

15 Diketahui : S = { x 10 < x 20, x B } M = { x x > 15, x S } N = { x x > 12,, x S } Gambarlah diagram vennya

16 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: S 11 N M

17 Algebra Law of Sets

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19 Set s Inclusion and Exclusion For A and B, Let A and B be any finite sets. Then : A B = A + B A B Inclusion Exclusion 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.

20 Set s Inclusion and Exclusion For A and B, Let A and B be any finite sets. Then : A B = A + B A B Inclusion Exclusion 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.

21 Inclusion and Exclusion of Sets For A and B, Let A and B be any finite sets. Then : A B = A + B A B Inclusion Exclusion 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.

22 Inclusion-Exclusion Principle 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}

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 c { (A B) c = { tidak suka keduanya} n((a B) c ) = 5 Maka A B = {suka keduanya} n(a B) = 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 5 n(s) = (20 x)+x+(46-x)+5 60 = 71 - x X = = 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 Berapa banyaknya bilangan bulat antara 1 dan 100 yang habis dibagi 3 atau 5?

25 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.