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.

1. Oleh
Arif Djunaidi

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 AB,
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. 푨 where U = {a, b, c, d} and A = {a}
b. 푩 where U = {1, 2, 3, . . .} and B = {2, 4, 6, .
. . }
c. 푨 ∪ 푩 and 푨 ∩ 푩 where U={1, 2, 3, 4, 5},
A={1, 2, 3}, and B= {3, 4}

19. SOLUTION
a. 푨 ={b, c, d}
b. 푩 = {1, 3, 5, . . . }
c. 푨 ∪ 푩={4, 5} ∪{1, 2, 5} = {1, 2, 4, 5}
푨 ∩ 푩 = 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)}