AKRAB BERSAMA SANDI MATEMATIKA
Oleh : Fithri Angelia Permana, S.Si (WI LPMP NAD)
Apakah matematika ilmu yang 'sulit'?
Secara umum, semakin kompleks suatu fenomena, semakin kompleks pula alat (dalam hal ini jenis matematika) yang melalui berbagai perumusan (model matematikanya) diharapkan mampu untuk mendapatkan atau sekedar mendekati solusi eksak seakurat-akuratnya.Jadi tingkat kesulitan suatu jenis atau cabang matematika bukan disebabkan oleh jenis atau cabang matematika itu sendiri, tetapi disebabkan oleh sulit dan kompleksnya fenomena yang solusinya diusahakan dicari atau didekati oleh perumusan (model matematikanya) dengan menggunakan jenis atau cabang matematika tersebut.
Sebaliknya berbagai fenomena fisik yg mudah di amati, misalnya jumlah penduduk di seluruh Indonesia, tak memerlukan jenis atau cabang matematika yang canggih. Kemampuan aritmatika sudah cukup untuk mencari solusi (jumlah penduduk) dengan keakuratan yang cukup tinggi.Dalam matematika sering digunakan simbol-simbol yang umum dikenal oleh matematikawan. Sering kali pengertian simbol ini tidak dijelaskan, karena dianggap maknanya telah diketahui. Hal ini kadang menyulitkan bagi mereka yang awam. Daftar berikut ini berisi banyak simbol beserta artinya.
Matematika sebagai bahasa
Di manakah letak semua konsep-konsep matematika, misalnya letak bilangan 1? Banyak para pakar matematika, misalnya para pakar Teori Model (lihat model matematika) yg juga mendalami filosofi di balik konsep-konsep matematika bersepakat bahwa semua konsep-konsep matematika secara universal terdapat di dalam pikiran setiap manusia.Jadi yang dipelajari dalam matematika adalah berbagai simbol dan ekspresi untuk mengkomunikasikannya. Misalnya orang Jawa secara lisan memberi simbol bilangan 3 dengan mengatakan "Telu", sedangkan dalam bahasa Indonesia, bilangan tersebut disimbolkan melalui ucapan "Tiga". Inilah sebabnya, banyak pakar mengkelompokkan matematika dalam kelompok bahasa, atau lebih umum lagi dalam kelompok (alat) komunikasi, bukan sains.
Dalam pandangan formalis, matematika adalah penelaahan struktur abstrak yang didefinisikan secara aksioma dengan menggunakan logika simbolik dan notasi matematika; ada pula pandangan lain, misalnya yang dibahas dalam filosofi matematika.
Struktur spesifik yang diselidiki oleh matematikawan sering kali berasal dari ilmu pengetahuan alam, dan sangat umum di fisika, tetapi matematikawan juga mendefinisikan dan menyelidiki struktur internal dalam matematika itu sendiri, misalnya, untuk menggeneralisasikan teori bagi beberapa sub-bidang, atau alat membantu untuk perhitungan biasa. Akhirnya, banyak matematikawan belajar bidang yang dilakukan mereka untuk sebab estetis saja, melihat ilmu pasti sebagai bentuk seni daripada sebagai ilmu praktis atau terapan.Matematika tingkat lanjut digunakan sebagai alat untuk mempelajari berbagai fenomena fisik yg kompleks, khususnya berbagai fenomena alam yang teramati, agar pola struktur, perubahan, ruang dan sifat-sifat fenomena bisa didekati atau dinyatakan dalam sebuah bentuk perumusan yg sistematis dan penuh dengan berbagai konvensi, simbol dan notasi. Hasil perumusan yang menggambarkan prilaku atau proses fenomena fisik tersebut biasa disebut model matematika dari fenomena.
Faktanya, kita melihat bahwa siswa kelas rendah (umumnya siswa kelas 1) lebih menyenangi pelajaran matematika dibanding pelajaran bahasa Indonesia. Hal ini, ”mungkin” karena matematika merupakan ilmu yang mudah dimengerti dengan lambang bilangan yang sederhana serta dekat dengan kehidupan si siswa. Fakta juga menunjukkan bahwa setelah siswa berada di kelas tinggi, mereka malah membenci matematika sampai kepada guru yang mengajarkannya. Banyak faktor yang mendukung terjadinya hal ini, guru yang kurang mampu mentransfer ilmunya, materi yang terlalu abstrak dan kode atau sandi di matematika yang tidak begitu familiar. Seseorang yang tidak paham sandi/kode dari lambang matematika akan mengalami kesulitan untuk memahami maksud permasalahan matematika yang ada. Ada beberapa sandi/kode matematika yang ditampilkan untuk menambah wawasan kita bersama.
Simbol matematika dasar mbol | Nama | Penjelasan | Contoh | | Dibaca sebagai | | | | Kategori | | | = | kesamaan | x = y berarti x and y mewakili hal atau nilai yang sama. | 1 + 1 = 2 | | sama dengan | | | | Umum | | | ≠ | Ketidaksamaan | x ≠ y berarti x dan y tidak mewakili hal atau nilai yang sama. | 1 ≠ 2 | | tidak sama dengan | | | | Umum | | | <

> | ketidaksamaan | x < y berarti x lebih kecil dari y.

x > y means x lebih besar dari y. | 3 < 4
5 > 4 | | lebih kecil dari; lebih besar dari | | | | order theory | | | ≤

≥ | inequality | x ≤ y berarti x lebih kecil dari atau sama dengan y.

x ≥ y means x lebih besar dari atau sama dengan y. | 3 ≤ 4 and 5 ≤ 5
5 ≥ 4 and 5 ≥ 5 | | lebih kecil dari atau sama dengan, lebih besar dari atau sama dengan | | | | order theory | | | + | tambah | 4 + 6 berarti jumlah antara 4 dan 6. | 2 + 7 = 9 | | tambah | | | | aritmatika | | | | disjoint union | A1 + A2 means the disjoint union of sets A1 and A2. | A1={1,2,3,4} ∧ A2={2,4,5,7} ⇒
A1 + A2 = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)} | | the disjoint union of … and … | | | | teori himpunan | | | − | kurang | 9 − 4 berarti 9 dikurangi 4. | 8 − 3 = 5 | | Kurang | | | | aritmatika | | | | tanda negatif | −3 berarti negatif dari angka 3. | −(−5) = 5 | | Negative | | | | aritmatika | | | | set-theoretic complement | A − B means the set that contains all the elements of A that are not in B. | {1,2,4} − {1,3,4} = {2} | | minus; without | | | | set theory | | | × | multiplication | 3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 | | Kali | | | | aritmatika | | | | Cartesian product | X×Y means the set of all ordered pairs with the first element of each pair selected from X and the second element selected from Y. | {1,2} × {3,4} = {(1,3),(1,4),(2,3),(2,4)} | | the Cartesian product of … and …; the direct product of … and … | | | | teori himpunan | | | | cross product | u × v means the cross product of vectors u and v | (1,2,5) × (3,4,−1) =
(−22, 16, − 2) | | Cross | | | | vector algebra | | | ÷/ | division | 6 ÷ 3 atau 6/3 berati 6 dibagi 3. | 2 ÷ 4 = .5

12/4 = 3 | | bagi | | | | aritmatika | | | √ | square root | √x berarti bilangan positif yang kuadratnya x. | √4 = 2 | | akar kuadrat | | | | bilangan real | | | | complex square root | if z = r exp(iφ) is represented in polar coordinates with -π < φ ≤ π, then √z = √r exp(iφ/2). | √(-1) = i | | the complex square root of; square root | | | | bilangan complex | | | | | | absolute value | |x| means the distance in the real line (or the complex plane) between x and zero. | |3| = 3, |-5| = |5|
|i| = 1, |3+4i| = 5 | | absolute value of | | | | numbers | | | ! | factorial | n! is the product 1×2×...×n. | 4! = 1 × 2 × 3 × 4 = 24 | | faktorial | | | | combinatorics | | | ~ | probability distribution | X ~ D, means the random variable X has the probability distribution D. | X ~ N(0,1), the standard normal distribution | | has distribution | | | | statistika | | | ⇒

→

⊃ | material implication | A ⇒ B means if A is true then B is also true; if A is false then nothing is said about B.

→ may mean the same as ⇒, or it may have the meaning for functions given below.

⊃ may mean the same as ⇒, or it may have the meaning for superset given below. | x = 2 ⇒ x2 = 4 is true, but x2 = 4 ⇒ x = 2 is in general false (since x could be −2). | | implies; if .. then | | | | propositional logic | | | ⇔

↔ | material equivalence | A ⇔ B means A is true if B is true and A is false if B is false. | x + 5 = y +2 ⇔ x + 3 = y | | if and only if; iff | | | | propositional logic | | | ¬

˜ | logical negation | The statement ¬A is true if and only if A is false.

A slash placed through another operator is the same as "¬" placed in front. | ¬(¬A) ⇔ A x ≠ y ⇔ ¬(x = y) | | not | | | | propositional logic | | | ∧ | logical conjunction or meet in a lattice | The statement A ∧ B is true if A and B are both true; else it is false. | n < 4 ∧ n >2 ⇔ n = 3 when n is a natural number. | | and | | | | propositional logic, lattice theory | | | ∨ | logical disjunction or join in a lattice | The statement A ∨ B is true if A or B (or both) are true; if both are false, the statement is false. | n ≥ 4 ∨ n ≤ 2 ⇔ n ≠ 3 when n is a natural number. | | or | | | | propositional logic, lattice theory | | | ⊕⊻ | exclusive or | The statement A ⊕ B is true when either A or B, but not both, are true. A ⊻ B means the same. | (¬A) ⊕ A is always true, A ⊕ A is always false. | | xor | | | | propositional logic, Boolean algebra | | | ∀ | universal quantification | ∀ x: P(x) means P(x) is true for all x. | ∀ n ∈ N: n2 ≥ n. | | for all; for any; for each | | | | predicate logic | | | ∃ | existential quantification | ∃ x: P(x) means there is at least one x such that P(x) is true. | ∃ n ∈ N: n is even. | | there exists | | | | predicate logic | | | ∃! | uniqueness quantification | ∃! x: P(x) means there is exactly one x such that P(x) is true. | ∃! n ∈ N: n + 5 = 2n. | | there exists exactly one | | | | predicate logic | | | :=

≡

:⇔ | definition | x := y or x ≡ y means x is defined to be another name for y (but note that ≡ can also mean other things, such as congruence).

P :⇔ Q means P is defined to be logically equivalent to Q. | cosh x := (1/2)(exp x + exp (−x))

A XOR B :⇔ (A ∨ B) ∧ ¬(A ∧ B) | | is defined as | | | | everywhere | | | { , } | set brackets | {a,b,c} means the set consisting of a, b, and c. | N = {0,1,2,...} | | the set of ... | | | | teori himpunan | | | { : }

{ | } | set builder notation | {x : P(x)} means the set of all x for which P(x) is true. {x | P(x)} is the same as {x : P(x)}. | {n ∈ N : n2 < 20} = {0,1,2,3,4} | | the set of ... such that ... | | | | teori himpunan | | | ∅

{} | himpunan kosong | ∅ berarti himpunan yang tidak memiliki elemen. {} juga berarti hal yang sama. | {n ∈ N : 1 < n2 < 4} = ∅ | | himpunan kosong | | | | teori himpunan | | | ∈

∉ | set membership | a ∈ S means a is an element of the set S; a ∉ S means a is not an element of S. | (1/2)−1 ∈ N

2−1 ∉ N | | is an element of; is not an element of | | | | everywhere, teori himpunan | | | ⊆

⊂ | subset | A ⊆ B means every element of A is also element of B.

A ⊂ B means A ⊆ B but A ≠ B. | A ∩ B ⊆ A; Q ⊂ R | | is a subset of | | | | teori himpunan | | | ⊇

⊃ | superset | A ⊇ B means every element of B is also element of A.

A ⊃ B means A ⊇ B but A ≠ B. | A ∪ B ⊇ B; R ⊃ Q | | is a superset of | | | | teori himpunan | | | ∪ | set-theoretic union | A ∪ B means the set that contains all the elements from A and also all those from B, but no others. | A ⊆ B ⇔ A ∪ B = B | | the union of ... and ...; union | | | | teori himpunan | | | ∩ | set-theoretic intersection | A ∩ B means the set that contains all those elements that A and B have in common. | {x ∈ R : x2 = 1} ∩ N = {1} | | intersected with; intersect | | | | teori himpunan | | | \ | set-theoretic complement | A \ B means the set that contains all those elements of A that are not in B. | {1,2,3,4} \ {3,4,5,6} = {1,2} | | minus; without | | | | teori himpunan | | | ( ) | function application | f(x) berarti nilai fungsi f pada elemen x. | Jika f(x) := x2, maka f(3) = 32 = 9. | | of | | | | teori himpunan | | | | precedence grouping | Perform the operations inside the parentheses first. | (8/4)/2 = 2/2 = 1, but 8/(4/2) = 8/2 = 4. | | | | | | umum | | | f:X→Y | function arrow | f: X → Y means the function f maps the set X into the set Y. | Let f: Z → N be defined by f(x) = x2. | | from ... to | | | | teori himpunan | | | O | function composition | fog is the function, such that (fog)(x) = f(g(x)). | if f(x) = 2x, and g(x) = x + 3, then (fog)(x) = 2(x + 3). | | composed with | | | | teori himpunan | | | Nℕ | natural numbers | N means {0,1,2,3,...}, but see the article on natural numbers for a different convention. | {|a| : a ∈ Z} = N | | N | | | | numbers | | | Zℤ | integers | Z means {...,−3,−2,−1,0,1,2,3,...}. | {a : |a| ∈ N} = Z | | Z | | | | numbers | | | Qℚ | rational numbers | Q means {p/q : p,q ∈ Z, q ≠ 0}. | 3.14 ∈ Q

π ∉ Q | | Q | | | | numbers | | | Rℝ | real numbers | R means {limn→∞ an : ∀ n ∈ N: an ∈ Q, the limit exists}. | π ∈ R

√(−1) ∉ R | | R | | | | numbers | | | Cℂ | complex numbers | C means {a + bi : a,b ∈ R}. | i = √(−1) ∈ C | | C | | | | numbers | | | ∞ | infinity | ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | limx→0 1/|x| = ∞ | | infinity | | | | numbers | | | Π | pi | π berarti perbandingan (rasio) antara keliling lingkaran dengan diameternya. | A = πr² adalah luas lingkaran dengan jari-jari (radius) r | | pi | | | | Euclidean geometry | | | || || | norm | ||x|| is the norm of the element x of a normed vector space. | ||x+y|| ≤ ||x|| + ||y|| | | norm of; length of | | | | linear algebra | | | ∑ | summation | ∑k=1n ak means a1 + a2 + ... + an. | ∑k=14 k2 = 12 + 22 + 32 + 42 = 1 + 4 + 9 + 16 = 30 | | sum over ... from ... to ... of | | | | aritmatika | | | ∏ | product | ∏k=1n ak means a1a2···an. | ∏k=14 (k + 2) = (1 + 2)(2 + 2)(3 + 2)(4 + 2) = 3 × 4 × 5 × 6 = 360 | | product over ... from ... to ... of | | | | aritmatika | | | | Cartesian product | ∏i=0nYi means the set of all (n+1)-tuples (y0,...,yn). | ∏n=13R = Rn | | the Cartesian product of; the direct product of | | | | set theory | | | ' | derivative | f '(x) is the derivative of the function f at the point x, i.e., the slope of the tangent there. | If f(x) = x2, then f '(x) = 2x | | … prime; derivative of … | | | | kalkulus | | | ∫ | indefinite integral or antiderivative | ∫ f(x) dx means a function whose derivative is f. | ∫x2 dx = x3/3 + C | | indefinite integral of …; the antiderivative of … | | | | kalkulus | | | | definite integral | ∫ab f(x) dx means the signed area between the x-axis and the graph of the function f between x = a and x = b. | ∫0b x2 dx = b3/3; | | integral from ... to ... of ... with respect to | | | | kalkulus | | | ∇ | gradient | ∇f (x1, …, xn) is the vector of partial derivatives (df / dx1, …, df / dxn). | If f (x,y,z) = 3xy + z² then ∇f = (3y, 3x, 2z) | | del, nabla, gradient of | | | | kalkulus | | | ∂ | partial derivative | With f (x1, …, xn), ∂f/∂xi is the derivative of f with respect to xi, with all other variables kept constant. | If f(x,y) = x2y, then ∂f/∂x = 2xy | | partial derivative of | | | | kalkulus | | | | boundary | ∂M means the boundary of M | ∂{x : ||x|| ≤ 2} =
{x : || x || = 2} | | boundary of | | | | topology | | | ⊥ | perpendicular | x ⊥ y means x is perpendicular to y; or more generally x is orthogonal to y. | If l⊥m and m⊥n then l || n. | | is perpendicular to | | | | geometri | | | | bottom element | x = ⊥ means x is the smallest element. | ∀x : x ∧ ⊥ = ⊥ | | the bottom element | | | | lattice theory | | | |= | entailment | A ⊧ B means the sentence A entails the sentence B, that is every model in which A is true, B is also true. | A ⊧ A ∨ ¬A | | entails | | | | model theory | | | |- | inference | x ⊢ y means y is derived from x. | A → B ⊢ ¬B → ¬A | | infers or is derived from | | | | propositional logic, predicate logic | | | ◅ | normal subgroup | N ◅ G means that N is a normal subgroup of group G. | Z(G) ◅ G | | is a normal subgroup of | | | | group theory | | | / | quotient group | G/H means the quotient of group G modulo its subgroup H. | {0, a, 2a, b, b+a, b+2a} / {0, b} = {{0, b}, {a, b+a}, {2a, b+2a}} | | mod | | | | group theory | | | ≈ | isomorphism | G ≈ H means that group G is isomorphic to group H | Q / {1, −1} ≈ V, where Q is the quaternion group and V is the Klein four-group. | | is isomorphic to | | | | group theory | | |
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