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Himpunan plpg

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The document defines and provides examples of key concepts in set theory, including: - A set is a collection of distinct objects called elements or members. Sets can be represented verbally or visually using lists or set-builder notation. - Two sets are equal if they contain the same elements. Operations on sets include union, intersection, complement, difference, and Cartesian product. - A set is finite if its elements can be counted, and infinite if its elements cannot be counted or listed with certainty. Examples demonstrate determining if a set is finite or infinite.

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Oleh Arif Djunaidi
Definisi Himpunan Sebuah koleksi benda-benda yang disebut Himpunan benda-benda yang disebut elemen, unsur atau anggota dari himpunan Deskripsi verbal "himpunan semua negara bagian di Amerika Serikat yang berbatasan dengan Samudera Pasifik" dapat direpresentasikan dalam dua cara berbeda sebagai berikut: Didaftar: {Alaska, California, Hawaii, Oregon, Washington}. Set-builder: {x/x adalah negara bagian AS yang berbatasan dengan Samudera Pasifik}. (Notasi set-builder ini dibaca: "Himpunan semua x sehingga x adalah negara bagian AS yang berbatasan dengan Samudera Pasifik.")
Himpunan biasanya dilambangkan dengan huruf kapital seperti A, B, C, dan seterusnya. Simbol " dan "" digunakan untuk menunjukkan bahwa sebuah objek atau bukan dari anggota himpunan. Misalnya, jika S merupakan himpunan semua negara bagian AS yang berbatasan dengan Pasifik, maka Alaska S dan Michigan S. Himpunan tanpa elemen disebut himpunan kosong (atau set null) dan dilambangkan dengan {} atau simbol . Himpunan semua negara bagian AS yang berbatasan Antartika adalah himpunan kosong.
Dua himpunan A dan B adalah sama, ditulis A = B, jika dan hanya jika mereka unsur-unsur sudah sesuai adalah sama. Jadi {x /x adalah negara yang berbatasan dengan Danau Michigan} = {Illinois, Indiana, Michigan, Wisconsin}. Perhatikan bahwa dua set, A dan B, yang sama jika setiap elemen A adalah di B, dan sebaliknya. Jika A tidak sama dengan B, kita menulis A = B. Ada dua aturan yang melekat mengenai set: (1) unsur yang sama tidak tercantum lebih dari sekali dalam satu set, dan (2) urutan unsur-unsur dalam satu set tidak material. Dengan demikian, berdasarkan aturan 1, set {a, a, b } akan ditulis sebagai {a, b} dan menurut aturan 2 {a, b} = {b, a}, {x, y, z} = {y, z, x}, dan seterusnya. Konsep 1-1 korespondensi, baca "satu-ke-satu korespondensi," diperlukan untuk menentukan arti seluruh bilangan.
DEFINISI One-to-One Correspondence Korespondensi satu satu antara dua set A dan B adalah pasangan dari elemen A dengan elemen B sehingga setiap elemen A berhubungan dengan tepat satu elemen dari B, dan sebaliknya. Jika ada 1-1 korespondensi antara set A dan B, kita menulis A B dan mengatakan bahwa A dan B adalah himpunan sama atau cocok.
example {1, 2} {a, b}, but {1, 2} {a, b}. The two sets A = {a, b} and B = {a, b, c} are not equivalent. However, they do satisfy the relationship defined next.
Definisi Subset of a Set: A B Set A is said to be a subset of B, written A B, if and only if every element of A is also an element of B. The set consisting of New Hampshire is a subset of the set of all New England states and {a, b, c} {a, b, c, d, e, f }. Since every element in a set A is in A, A A for all sets A. Also, {a, b, c} {a, b, d} because c is in the set {a, b, c} but not in the set {a, b, d}. Using similar reasoning, you can argue that A for any set A since it is impossible to find an element in that is not in A.
If A B and B has an element that is not in A, we write A B and say that A is a proper subset of B. Thus {a, b} {a, b, c}, since {a, b} {a, b, c} and c is in the second set but not in the first.
Circles or other closed curves are used in Venn diagrams (named after the English logician John Venn) to illustrate relationships between sets. These circles are usually pictured within a rectangle, U, where the rectangle represents the universal set or universe
Finite and Infinite Sets a set is finite if it is empty or can have its elements listed (where the list eventually ends), and a set is infinite if it goes on without end. Example: Determine whether the following sets are finite or infinite. a. {a, b, c} b. {1, 2, 3, . . . } c. {2, 4, 6, . . . , 2 0 }
SOLUTION a. {a, b, c} is finite since it can be matched with the set {1, 2, 3}. b. {1, 2, 3, . . .} is an infinite set. c. {2, 4, 6, . . . , 20} is a finite set since it can be matched with the set {1, 2, 3, . . . ,10}. (Here, the ellipsis means to continue the pattern until the last element is reached.)
Operations on Sets Two sets A and B that have no elements in common are called disjoint sets. The sets {a, b, c} and {d, e, f } are disjoint
Union of Sets: A B The union of two sets A and B, written A B, is the set that consists of all elements belonging either to A or to B (or to both). Example: Find the union of the given pairs of sets. a. {a, b} {c, d, e} b. {1, 2, 3, 4, 5} c. {m, n, q} {m, n, p}
SOLUTION a. {a, b} {c, d, e} = {a, b, c, d, e} b. {1, 2, 3, 4, 5} ={1, 2, 3, 4, 5} c. {m, n, q} {m, n, p} = {m, n, p, q}
Intersection of Sets: A B The intersection of sets A and B, written AB, is the set of all elements common to sets A and B. Example: Find the intersection of the given pairs of sets. a. {a, b, c} {b, d, f } b. {a, b, c} {a, b, c} c. {a, b} {c, d}
SOLUTION a. {a, b, c} {b, d, f} = {b} since b is the only element in both sets. b. {a, b, c} {a, b, c} = {a, b, c} since a, b, c are in both sets. c. {a, b} {c, d} = since there are no elements common to the given two sets.
Complement of a Set: The complement of a set A, written , is the set of all elements in the universe, U, that are not in A.
Example: Find the following sets. a. where U = {a, b, c, d} and A = {a} b. where U = {1, 2, 3, . . .} and B = {2, 4, 6, . . . } c. and where U={1, 2, 3, 4, 5}, A={1, 2, 3}, and B= {3, 4}
SOLUTION a. ={b, c, d} b. = {1, 3, 5, . . . } c. ={4, 5} {1, 2, 5} = {1, 2, 4, 5} = 3 ={1, 2, 4, 5}
Difference of Sets: A B The set difference (or relative complement) of set B from set A, written A B, is the set of all elements in A that are not in B.
Example: Find the difference of the given pairs of sets. a. {a, b, c} - {b, d} b. {a, b, c} - {e} c. {a, b, c, d} - {b, c, d}
SOLUTION a. {a, b, c} - {b, d} = {a, c} b. {a, b, c} - {e} = {a, b, c} c. {a, b, c, d} - {b, c, d} = {a}
Cartesian Product of Sets: A B The Cartesian product of set A with set B, written A B and read A cross B, is the set of all ordered pairs (a, b), where a A and b B. Example: Find the Cartesian product of the given pairs of sets. a. {x, y, z} {m, n} b. {7} {a, b, c}
SOLUTION a. {x, y, z} {m, n} = {(x, m), (x, n), (y, m), (y, n), (z, m), (z, n)} b. {7} {a, b, c} = {(7, a), (7, b), (7, c)}
Himpunan plpg
Himpunan plpg
Himpunan plpg

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Himpunan plpg

  • 2. Definisi Himpunan Sebuah koleksi benda-benda yang disebut Himpunan benda-benda yang disebut elemen, unsur atau anggota dari himpunan Deskripsi verbal "himpunan semua negara bagian di Amerika Serikat yang berbatasan dengan Samudera Pasifik" dapat direpresentasikan dalam dua cara berbeda sebagai berikut: Didaftar: {Alaska, California, Hawaii, Oregon, Washington}. Set-builder: {x/x adalah negara bagian AS yang berbatasan dengan Samudera Pasifik}. (Notasi set-builder ini dibaca: "Himpunan semua x sehingga x adalah negara bagian AS yang berbatasan dengan Samudera Pasifik.")
  • 3. Himpunan biasanya dilambangkan dengan huruf kapital seperti A, B, C, dan seterusnya. Simbol " dan "" digunakan untuk menunjukkan bahwa sebuah objek atau bukan dari anggota himpunan. Misalnya, jika S merupakan himpunan semua negara bagian AS yang berbatasan dengan Pasifik, maka Alaska S dan Michigan S. Himpunan tanpa elemen disebut himpunan kosong (atau set null) dan dilambangkan dengan {} atau simbol . Himpunan semua negara bagian AS yang berbatasan Antartika adalah himpunan kosong.
  • 4. Dua himpunan A dan B adalah sama, ditulis A = B, jika dan hanya jika mereka unsur-unsur sudah sesuai adalah sama. Jadi {x /x adalah negara yang berbatasan dengan Danau Michigan} = {Illinois, Indiana, Michigan, Wisconsin}. Perhatikan bahwa dua set, A dan B, yang sama jika setiap elemen A adalah di B, dan sebaliknya. Jika A tidak sama dengan B, kita menulis A = B. Ada dua aturan yang melekat mengenai set: (1) unsur yang sama tidak tercantum lebih dari sekali dalam satu set, dan (2) urutan unsur-unsur dalam satu set tidak material. Dengan demikian, berdasarkan aturan 1, set {a, a, b } akan ditulis sebagai {a, b} dan menurut aturan 2 {a, b} = {b, a}, {x, y, z} = {y, z, x}, dan seterusnya. Konsep 1-1 korespondensi, baca "satu-ke-satu korespondensi," diperlukan untuk menentukan arti seluruh bilangan.
  • 5. DEFINISI One-to-One Correspondence Korespondensi satu satu antara dua set A dan B adalah pasangan dari elemen A dengan elemen B sehingga setiap elemen A berhubungan dengan tepat satu elemen dari B, dan sebaliknya. Jika ada 1-1 korespondensi antara set A dan B, kita menulis A B dan mengatakan bahwa A dan B adalah himpunan sama atau cocok.
  • 6. example {1, 2} {a, b}, but {1, 2} {a, b}. The two sets A = {a, b} and B = {a, b, c} are not equivalent. However, they do satisfy the relationship defined next.
  • 7. Definisi Subset of a Set: A B Set A is said to be a subset of B, written A B, if and only if every element of A is also an element of B. The set consisting of New Hampshire is a subset of the set of all New England states and {a, b, c} {a, b, c, d, e, f }. Since every element in a set A is in A, A A for all sets A. Also, {a, b, c} {a, b, d} because c is in the set {a, b, c} but not in the set {a, b, d}. Using similar reasoning, you can argue that A for any set A since it is impossible to find an element in that is not in A.
  • 8. If A B and B has an element that is not in A, we write A B and say that A is a proper subset of B. Thus {a, b} {a, b, c}, since {a, b} {a, b, c} and c is in the second set but not in the first.
  • 9. Circles or other closed curves are used in Venn diagrams (named after the English logician John Venn) to illustrate relationships between sets. These circles are usually pictured within a rectangle, U, where the rectangle represents the universal set or universe
  • 10. Finite and Infinite Sets a set is finite if it is empty or can have its elements listed (where the list eventually ends), and a set is infinite if it goes on without end. Example: Determine whether the following sets are finite or infinite. a. {a, b, c} b. {1, 2, 3, . . . } c. {2, 4, 6, . . . , 2 0 }
  • 11. SOLUTION a. {a, b, c} is finite since it can be matched with the set {1, 2, 3}. b. {1, 2, 3, . . .} is an infinite set. c. {2, 4, 6, . . . , 20} is a finite set since it can be matched with the set {1, 2, 3, . . . ,10}. (Here, the ellipsis means to continue the pattern until the last element is reached.)
  • 12. Operations on Sets Two sets A and B that have no elements in common are called disjoint sets. The sets {a, b, c} and {d, e, f } are disjoint
  • 13. Union of Sets: A B The union of two sets A and B, written A B, is the set that consists of all elements belonging either to A or to B (or to both). Example: Find the union of the given pairs of sets. a. {a, b} {c, d, e} b. {1, 2, 3, 4, 5} c. {m, n, q} {m, n, p}
  • 14. SOLUTION a. {a, b} {c, d, e} = {a, b, c, d, e} b. {1, 2, 3, 4, 5} ={1, 2, 3, 4, 5} c. {m, n, q} {m, n, p} = {m, n, p, q}
  • 15. Intersection of Sets: A B The intersection of sets A and B, written AB, is the set of all elements common to sets A and B. Example: Find the intersection of the given pairs of sets. a. {a, b, c} {b, d, f } b. {a, b, c} {a, b, c} c. {a, b} {c, d}
  • 16. SOLUTION a. {a, b, c} {b, d, f} = {b} since b is the only element in both sets. b. {a, b, c} {a, b, c} = {a, b, c} since a, b, c are in both sets. c. {a, b} {c, d} = since there are no elements common to the given two sets.
  • 17. Complement of a Set: The complement of a set A, written , is the set of all elements in the universe, U, that are not in A.
  • 18. Example: Find the following sets. a. where U = {a, b, c, d} and A = {a} b. where U = {1, 2, 3, . . .} and B = {2, 4, 6, . . . } c. and where U={1, 2, 3, 4, 5}, A={1, 2, 3}, and B= {3, 4}
  • 19. SOLUTION a. ={b, c, d} b. = {1, 3, 5, . . . } c. ={4, 5} {1, 2, 5} = {1, 2, 4, 5} = 3 ={1, 2, 4, 5}
  • 20. Difference of Sets: A B The set difference (or relative complement) of set B from set A, written A B, is the set of all elements in A that are not in B.
  • 21. Example: Find the difference of the given pairs of sets. a. {a, b, c} - {b, d} b. {a, b, c} - {e} c. {a, b, c, d} - {b, c, d}
  • 22. SOLUTION a. {a, b, c} - {b, d} = {a, c} b. {a, b, c} - {e} = {a, b, c} c. {a, b, c, d} - {b, c, d} = {a}
  • 23. Cartesian Product of Sets: A B The Cartesian product of set A with set B, written A B and read A cross B, is the set of all ordered pairs (a, b), where a A and b B. Example: Find the Cartesian product of the given pairs of sets. a. {x, y, z} {m, n} b. {7} {a, b, c}
  • 24. SOLUTION a. {x, y, z} {m, n} = {(x, m), (x, n), (y, m), (y, n), (z, m), (z, n)} b. {7} {a, b, c} = {(7, a), (7, b), (7, c)}