# கணிதக் குறியீடுகள்


கணித தகவல்களை வெளிப்படுத்த **கணிதக் குறியீடுகள்** பயன்படுகின்றன. எழுத்துக்கள் எப்படி மொழியூடாக தகவல்களை வெளிப்படுத்த அவசியமோ அதேபோல் குறியீடுகள் கணிதத்தினூடாக தகவல்களை வெளிப்படுத்த அவசியம். ஒரு மொழியை அறிய, பயன்படுத்த எப்படி அதன் எழுத்துக்களை அறிவது அவசியமோ அதேபோல் கணிதத்தை அறிய, பயன்படுத்த கணிதக் குறியீடுகளை அறிவது அவசியம். எண்கள், செயற்பாட்டுக் குறியீடுகள், கருத்துருக் குறியீடுகள், சமன்பாடுகள் என பலநிலையிலான குறியீடுகள் கணிதத்தில் உண்டு.  

## அடிப்படை கணிதக் குறியீடுகள் அட்டவணை

| பலுக்கும் முறை |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- |
| rowspan=3 bgcolor=#d0f0d0 align=<div>=</div> | சமம் | காட்டாக 2 + 3 = 5 என்பது ஒரு சமன்பாடு. இதனை 2 கூட்டல் 3 **ஈடு** 5 என்று படிக்கலாம், அல்லது 2 கூட்டல் 3 **சமம்** 5 என்று படிக்கலாம். அதே போல 2 + 4 = 3 x 2 என்பதும் ஒரு சமன்பாடு. | 1&nbsp;+ 1&nbsp;= 2 |  |  |
| align=சமமாக, ஈடாக |  |  |  |  |  |
| align=எங்கும் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>≠

<>

!=</div> | சமனிலி | *x*&nbsp;≠ *y* என்பது *x* ம் *y* யும் ஒன்றல்ல, ஒரே மதிப்பைக் கொள்ளவில்லை. .

(*குறியீடுகள்* != *ம்* <> '' கணினியியலில் பயன்படுகிறது.) | 1 ≠ 2 |  |  |
| align=சமமில்லை |  |  |  |  |  |
| align= valign= |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div><

>

≪

≫</div> | strict inequality | *x*&nbsp;< *y* என்பது *x* ஐவிடச் சிறியது *y*.

*x*&nbsp;> *y* என்பது *x* *y*யிலும் பெரியது.

*x*&nbsp;≪ *y* என்பது *x* *y* ஐவிட மிகச் சிறியது.

*x*&nbsp;≫ *y* என்பது *x* *y*ஐவிடப் மிகவும் பெரியது. | 3&nbsp;< 4
5&nbsp;> 4.

0.003&nbsp;≪ 1000000 |  |  |
| align=is less than, is greater than, is much less than, is much greater than |  |  |  |  |  |
| align=order theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>≤
<=

≥
>=</div> | inequality | *x*&nbsp;≤ *y* means *x* is less than or equal to *y*.

*x*&nbsp;≥ *y* means *x* is greater than or equal to *y*.

(*The symbols* <= *and* >= *are primarily from computer science. They are avoided in mathematical texts.*) | 3&nbsp;≤&nbsp;4 and 5&nbsp;≤&nbsp;5
5&nbsp;≥&nbsp;4 and 5&nbsp;≥&nbsp;5 |  |  |
| align=is less than or equal to, is greater than or equal to |  |  |  |  |  |
| align=order theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∝</div> | proportionality | *y* ∝ *x* means that *y* = *kx* for some constant *k*. | if *y* = 2*x*, then *y* ∝ *x* |  |  |
| align=is proportional to; varies as |  |  |  |  |  |
| align= valign=everywhere |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>+</div> | கூட்டல் | 4 + 6 means the sum of 4 and 6. | 2 + 7 = 9 |  |  |
| align=plus |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| disjoint union | *A*<sub>1</sub> + *A*<sub>2</sub> means the disjoint union of sets *A*<sub>1</sub> and *A*<sub>2</sub>. | *A*<sub>1</sub> = {1, 2, 3, 4} ∧ *A*<sub>2</sub> = {2, 4, 5, 7} ⇒
*A*<sub>1</sub> + *A*<sub>2</sub> = {(1,1), (2,1), (3,1), (4,1), (2,2), (4,2), (5,2), (7,2)} |  |  |  |
| align=the disjoint union of … and … |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=9 bgcolor=#d0f0d0 align=<div>−</div> | கழித்தல் | 9 − 4 means the subtraction of 4 from 9. | 8 − 3 = 5 |  |  |
| align=minus |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| negative sign | −3 means the negative of the number 3. | −(−5) = 5 |  |  |  |
| align=negative; minus |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| set-theoretic complement | *A*&nbsp;−&nbsp;*B* means the set that contains all the elements of *A* that are not in *B*.

∖ can also be used for set-theoretic complement as described below. | {1,2,4}&nbsp;−&nbsp;{1,3,4}&nbsp;&nbsp;=&nbsp; {2} |  |  |  |
| align=minus; without |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=9 bgcolor=#d0f0d0 align=<div>×</div> | பெருக்கல் | 3 × 4 means the multiplication of 3 by 4. | 7 × 8 = 56 |  |  |
| align=times |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| கார்ட்டீசியன் பெருக்கற்பலன் | *X*×*Y* means the set of all வரிசைச் சோடி 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)} |  |  |  |
| align=the Cartesian product of … and ...; the direct product of … and … |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| குறுக்குப் பெருக்கு | **u** × **v** means the cross product of vectors **u** and **v** | (1,2,5) × (3,4,−1) =
(−22, 16, − 2) |  |  |  |
| align=cross |  |  |  |  |  |
| align=vector algebra |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>·</div> | பெருக்கல் | 3 · 4 means the multiplication of 3 by 4. | 7 · 8 = 56 |  |  |
| align=times |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| புள்ளிப் பெருக்கல் | **u** · **v** means the dot product of vectors **u** and **v** | (1,2,5) · (3,4,−1) = 6 |  |  |  |
| align=dot |  |  |  |  |  |
| align=vector algebra |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>÷

&frasl;</div> | division | 6 ÷ 3 or 6 &frasl; 3 means the division of 6 by 3. | 2 ÷ 4 = .5

12 &frasl; 4 = 3 |  |  |
| align=divided by |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>±</div> | plus-minus | 6 ± 3 means both 6 + 3 and 6 – 3. | The equation *x* = 5 ± √4, has two solutions, *x* = 7 and *x* = 3. |  |  |
| align=plus or minus |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| plus-minus | 10 ± 2 or equivalently 10 ± 20% means the range from 10 − 2 to 10 + 2. | If *a* = 100 ± 1 mm, then *a* ≥ 99&nbsp;mm and *a* ≤ 101&nbsp;mm. |  |  |  |
| align=plus or minus |  |  |  |  |  |
| align=அளவியல் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | minus-plus | 6 ± (3 5) means both 6 + (3 – 5) and 6 – (3 + 5). | cos(*x* ± *y*) = cos(*x*) cos(*y*) sin(*x*) sin(*y*). |  |  |
| align=minus or plus |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>√</div> | வர்க்கமூலம் | √*x* means the positive number whose square is *x*. | √4&nbsp;= 2 |  |  |
| align=the principal square root of; square root |  |  |  |  |  |
| align=மெய்யெண் |  |  |  |  |  |
| complex square root | if *z* = *r* exp(***i***&phi;) is represented in polar coordinates with -***π*** < &phi; ≤ ***π***, then √*z* = √*r* exp(***i***&nbsp;&phi;/2). | √(-1)&nbsp;= ***i*** |  |  |  |
| align=the complex square root of&nbsp;…

square root |  |  |  |  |  |
| align=சிக்கலெண் |  |  |  |  |  |
| </div> | தனி மதிப்பு or modulus | means the distance along the real line (or across the complex plane) between *x* and zero. | &nbsp;= 3 | = 1 | &nbsp;= 5 |
| align=absolute value (modulus) of |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| யூக்ளிடிய தொலைவு | means the Euclidean distance between **x** and **y**. | For **x**&nbsp;= (1,1), and **y**&nbsp;= (4,5), | = √([1–4]<sup>2</sup> + [1–5]<sup>2</sup>)&nbsp;= 5 |  |  |
| align=Euclidean distance between; Euclidean norm of |  |  |  |  |  |
| align=வடிவவியல் |  |  |  |  |  |
| அணிக்கோவை | means the determinant of the matrix **A** | <math>\begin{vmatrix}
1&2 \\
2&4 \\
\end{vmatrix} = 0</math> |  |  |  |
| align=determinant of |  |  |  |  |  |
| align=அணி (கணிதம்) |  |  |  |  |  |
| </div> | divides | *b* means *a* divides *b*. | 15. |  |  |
| align=divides |  |  |  |  |  |
| align=Number Theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>!</div> | தொடர் பெருக்கம் | *n*&nbsp;**!** is the product 1 × 2× … × *n*. | 4**!** = 1 × 2 × 3 × 4 = 24 |  |  |
| align=factorial |  |  |  |  |  |
| align=சேர்வியல் (கணிதம்) |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>T</div> | transpose | Swap rows for columns | <math>A_{ij} = (A^T)_{ji}</math> |  |  |
| align=transpose |  |  |  |  |  |
| align=அணி (கணிதம்)s |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>~</div> | நிகழ்தகவுப் பரவல் | *X ~ D*, means the சமவாய்ப்பு மாறி *X* has the probability distribution *D*. | ''X ~ N(0,1), the இயல்நிலைப் பரவல் |  |  |
| align=has distribution |  |  |  |  |  |
| align=புள்ளியியல் |  |  |  |  |  |
| Row equivalence | *A*~*B* means that *B* can be generated by using a series of elementary row operations on *A* | <math>\begin{bmatrix}
1&2 \\
2&4 \\
\end{bmatrix} \sim \begin{bmatrix}
1&2 \\
0&0 \\
\end{bmatrix}</math> |  |  |  |
| align=is row equivalent to |  |  |  |  |  |
| align=அணி (கணிதம்) |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⇒

→

⊃</div> | 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&nbsp;&nbsp;⇒&nbsp; *x*<sup>2</sup> = 4 is true, but *x*<sup>2</sup> = 4 &nbsp;&nbsp;⇒&nbsp; *x* = 2 is in general false (since *x* could be −2). |  |  |
| align=implies; if … then |  |  |  |  |  |
| align=propositional logic, Heyting algebra |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⇔

↔</div> | material equivalence | *A*&nbsp;⇔ *B* means *A* is true if *B* is true and *A* is false if *B* is false. | *x*&nbsp;+ 5&nbsp;= *y*&nbsp;+2&nbsp;&nbsp;⇔&nbsp; *x*&nbsp;+ 3&nbsp;= *y* |  |  |
| align=if and only if; iff |  |  |  |  |  |
| align=propositional logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>¬

˜</div> | 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.

(*The symbol* ~ *has many other uses, so* ¬ * or the slash notation is preferred.*) | ¬(¬*A*)&nbsp;⇔ *A*
*x*&nbsp;≠&nbsp;*y*&nbsp;&nbsp;⇔&nbsp; ¬(*x*&nbsp;=&nbsp; *y*) |  |  |
| align=not |  |  |  |  |  |
| align=propositional logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∧</div> | logical conjunction or **meet** in a lattice | The statement *A* ∧ *B* is true if *A* and *B* are both true; else it is false.

For functions *A*(x) and *B*(x), *A*(x) ∧ *B*(x) is used to mean min(A(x), B(x)). | *n*&nbsp;< 4&nbsp;&nbsp;∧&nbsp; *n*&nbsp;>2&nbsp;&nbsp;⇔&nbsp; *n*&nbsp;= 3 when *n* is a இயல் எண். |  |  |
| align=and; min |  |  |  |  |  |
| align=propositional logic, lattice theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∨</div> | 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.

For functions *A*(x) and *B*(x), *A*(x) ∨ *B*(x) is used to mean max(A(x), B(x)). | *n*&nbsp;≥ 4&nbsp;&nbsp;∨&nbsp; *n*&nbsp;≤ 2&nbsp;&nbsp;⇔ *n*&nbsp;≠ 3 when *n* is a இயல் எண். |  |  |
| align=or; max |  |  |  |  |  |
| align=propositional logic, lattice theory |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=
<div>⊕</div>

<div></div> | 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. |  |  |
| align=xor |  |  |  |  |  |
| align=propositional logic, Boolean algebra |  |  |  |  |  |
| direct sum | The direct sum is a special way of combining several modules into one general module (the symbol ⊕ is used, is only for logic). | Most commonly, for vector spaces *U*, *V*, and *W*, the following consequence is used:
*U* = *V* ⊕ *W* ⇔ (*U* = *V* + *W*) ∧ (*V* ∩ *W* = ) |  |  |  |
| align=direct sum of |  |  |  |  |  |
| align=Abstract algebra |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&forall;</div> | universal quantification | &forall;&nbsp;*x*: *P*(*x*) means *P*(*x*) is true for all *x*. | &forall;&nbsp;*n*&nbsp;∈&nbsp;: *n*<sup>2</sup>&nbsp;≥ *n*. |  |  |
| align=for all; for any; for each |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&exist;</div> | existential quantification | &exist;&nbsp;*x*: *P*(*x*) means there is at least one *x* such that *P*(*x*) is true. | &exist;&nbsp;*n*&nbsp;∈&nbsp;: *n* is even. |  |  |
| align=there exists |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>&exist;!</div> | uniqueness quantification | &exist;!&nbsp;*x*: *P*(*x*) means there is exactly one *x* such that *P*(*x*) is true. | &exist;!&nbsp;*n*&nbsp;∈&nbsp;: *n*&nbsp;+ 5&nbsp;= 2*n*. |  |  |
| align=there exists exactly one |  |  |  |  |  |
| align=predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>:=

≡

:⇔</div> | வரைவிலக்கணம் | *x*&nbsp;:= *y* or *x*&nbsp;≡ *y* means *x* is defined to be another name for *y*

(*Some writers use* ≡ *to mean congruence*).

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

*A*&nbsp;**xor**&nbsp;*B* :⇔ (*A*&nbsp;∨&nbsp;*B*)&nbsp;∧&nbsp;¬(*A*&nbsp;∧&nbsp;*B*) |  |  |
| align=is defined as |  |  |  |  |  |
| align=everywhere |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | congruence | △ABC △DEF means triangle ABC is congruent to (has the same measurements as) triangle DEF. |  |  |  |
| align=is congruent to |  |  |  |  |  |
| align=வடிவவியல் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>≡</div> | congruence relation | a ≡ b (mod n) means a − b is divisible by n | 5 ≡ 11 (mod 3) |  |  |
| align=… is congruent to … modulo … |  |  |  |  |  |
| align=சமானம், மாடுலோ n |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>{&nbsp;,&nbsp;}</div> | தொடை brackets | {*a*,*b*,*c*} means the set consisting of *a*, *b*, and *c*. | &nbsp;= {&nbsp;1, 2, 3, …} |  |  |
| align=the set of … |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| &nbsp;}</div> | set builder notation | *P*(*x*)} is the same as {*x*&nbsp;: *P*(*x*)}. | {*n*&nbsp;∈&nbsp;&nbsp;: *n*<sup>2</sup>&nbsp;<&nbsp;20}&nbsp;= {&nbsp;1, 2, 3, 4} |  |  |
| align=the set of … such that |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

{&nbsp;}</div> | சூனியத்தொடை | means the set with no elements. {&nbsp;} means the same. | {*n*&nbsp;∈&nbsp;&nbsp;: 1&nbsp;<&nbsp;*n*<sup>2</sup>&nbsp;<&nbsp;4}&nbsp;= |  |  |
| align= the empty set |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∈

</div> | set membership | *a*&nbsp;∈ *S* என்பது *a* , *S*தொடையின் மூலகமாகும் ; *a*&nbsp; *S* என்பது *a* ,*S*தொடையின் மூலகமல்ல என்றும் குறித்து நிற்கும் . | (1/2)<sup>−1</sup>&nbsp;∈&nbsp;

2<sup>−1</sup>&nbsp;&nbsp; |  |  |
| align=மூலகம் ; மூலகமன்று |  |  |  |  |  |
| align=everywhere, set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⊆

⊂</div> | உபதொடை | (subset) *A*&nbsp;⊆&nbsp;*B* means every element of *A* is also element of *B*.

(proper subset) *A*&nbsp;⊂&nbsp;*B* means *A*&nbsp;⊆&nbsp;*B* but *A*&nbsp;≠&nbsp;*B*.

(*Some writers use the symbol ⊂ as if it were the same as ⊆.*) | (*A*&nbsp;∩&nbsp;*B*)&nbsp;⊆&nbsp;*A*

&nbsp;⊂&nbsp;

&nbsp;⊂&nbsp; |  |  |
| align=is a subset of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>⊇

⊃</div> | superset | *A*&nbsp;⊇&nbsp;*B* means every element of *B* is also element of *A*.

*A*&nbsp;⊃&nbsp;*B* means *A*&nbsp;⊇&nbsp;*B* but *A*&nbsp;≠&nbsp;*B*.

(*Some writers use the symbol ⊃ as if it were the same as ⊇.*) | (*A*&nbsp;∪&nbsp;*B*)&nbsp;⊇&nbsp;*B*

&nbsp;⊃&nbsp; |  |  |
| align=is a superset of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∪</div> | set-theoretic union | (exclusive) *A*&nbsp;∪&nbsp;*B* means the set that contains all the elements from *A*, or all the elements from *B*, but not both.
"*A* or *B*, but not both."

(inclusive) *A*&nbsp;∪&nbsp;*B* means the set that contains all the elements from *A*, or all the elements from *B*, or all the elements from both *A* and *B*.
"*A* or *B* or both". | *A*&nbsp;⊆&nbsp;*B*&nbsp;&nbsp;⇔&nbsp; (*A*&nbsp;∪&nbsp;*B*)&nbsp;=&nbsp;*B* (inclusive) |  |  |
| align=the union of … and …

union |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∩</div> | set-theoretic intersection | *A*&nbsp;∩&nbsp;*B* means the set that contains all those elements that *A* and *B* have in common. | {*x*&nbsp;∈&nbsp;&nbsp;: *x*<sup>2</sup>&nbsp;= 1}&nbsp;∩&nbsp;&nbsp;= {1} |  |  |
| align=intersected with; intersect |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div> <math>\Delta</math> </div> | symmetric difference | <math> A\Delta B</math> means the set of elements in exactly one of *A* or *B*. | {1,5,6,8} <math>\Delta</math> {2,5,8} = {1,2,6} |  |  |
| align=symmetric difference |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | set-theoretic complement | *A*&nbsp;&nbsp;*B* means the set that contains all those elements of *A* that are not in *B*.

− can also be used for set-theoretic complement as described above. | {1,2,3,4}&nbsp;&nbsp;{3,4,5,6}&nbsp;= {1,2} |  |  |
| align=minus; without |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>(&nbsp;)</div> | function application | *f*(*x*) means the value of the function *f* at the element *x*. | If *f*(*x*)&nbsp;:= *x*<sup>2</sup>, then *f*(3)&nbsp;= 3<sup>2</sup>&nbsp;= 9. |  |  |
| align=of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| precedence grouping | Perform the operations inside the parentheses first. | (8/4)/2&nbsp;= 2/2&nbsp;= 1, but 8/(4/2)&nbsp;= 8/2&nbsp;= 4. |  |  |  |
| align=parentheses |  |  |  |  |  |
| align=everywhere |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>*f*:*X*→*Y*</div> | function arrow | *f*:&nbsp;*X*&nbsp;→ *Y* means the function *f* maps the set *X* into the set *Y*. | Let *f*:&nbsp;&nbsp;→&nbsp; be defined by *f*(*x*)&nbsp;:= *x*<sup>2</sup>. |  |  |
| align=from … to |  |  |  |  |  |
| align=set theory,type theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div><small>o</small></div> | சார்புகளின் தொகுப்பு | *f*<small>o</small>*g* is the function, such that (*f*<small>o</small>*g*)(*x*) = *f*(*g*(*x*)). | if *f*(*x*) := 2*x*, and *g*(*x*) := *x* + 3, then (*f*<small>o</small>*g*)(*x*) = 2(*x* + 3). |  |  |
| align=composed with |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

**N**</div> | இயற்கை எண்கள் | **N** means {&nbsp;1, 2, 3, …}, but see the article on natural numbers for a different convention. | &nbsp;: *a*&nbsp;∈ , *a*&nbsp;≠ 0} |  |  |
| align=N |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

**Z**</div> | நிறை எண்கள் | means {..., −3, −2, −1, 0, 1, 2, 3, …} and <sup>+</sup> means {1, 2, 3, …}&nbsp;= . | &nbsp;= {*p*,&nbsp;-*p*&nbsp;: *p*&nbsp;∈ }&nbsp;∪&nbsp;{0} |  |  |
| align=Z |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

**Q**</div> | விகிதமுறு எண்கள் | means {*p*/*q*&nbsp;: *p*&nbsp;∈&nbsp;, *q*&nbsp;∈&nbsp;}. | 3.14000...&nbsp;∈

π&nbsp; |  |  |
| align=Q |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

**R**</div> | மெய்யெண்s | ℝ means the set of real numbers. | π&nbsp;∈

√(−1)&nbsp; |  |  |
| align=R |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>

**C**</div> | சிக்கலெண்s | means {*a*&nbsp;+&nbsp;*b*&nbsp;***i***&nbsp;: *a*,*b*&nbsp;∈&nbsp;}. | ***i***&nbsp;= √(−1)&nbsp;∈ |  |  |
| C'' |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| arbitrary constant | *C* can be any number, most likely unknown; usually occurs when calculating antiderivatives. | if *f(x)* = 6*x*² + 4*x*, then *F(x)* = 2*x*³ + 2*x*² + *C*, where *F'(x)* = *f(x)* |  |  |  |
| align=C |  |  |  |  |  |
| align=தொகையீடு |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>

**K**</div> | real or சிக்கலெண்s | **K** means the statement holds substituting **K** for **R** and also for **C**. | : <math>x^2\in\mathbb{C}\,\forall x\in \mathbb{K}</math>
because
: <math>x^2\in\mathbb{C}\,\forall x\in \mathbb{R}</math>
and
:<math>x^2\in\mathbb{C}\,\forall x\in \mathbb{C}</math>. |  |  |
| align=K |  |  |  |  |  |
| align=நேரியல் இயற்கணிதம் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∞</div> | எண்ணிலி | ∞ is an element of the extended number line that is greater than all real numbers; it often occurs in limits. | } = \infty</math> |  |  |
| align=எண்ணிலி |  |  |  |  |  |
| align=எண்s |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>…</div> | norm | &nbsp;*x*&nbsp; is the norm of the element *x* of a normed vector space. | &nbsp;*x*&nbsp; + *y*&nbsp; ≤&nbsp; &nbsp;*x*&nbsp;&nbsp; +&nbsp; &nbsp;*y*&nbsp; |  |  |
| align=norm of

length of |  |  |  |  |  |
| align= நேரியல் இயற்கணிதம் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∑</div> | summation | <math>\sum_{k=1}^{n}{a_k}</math> means *a*<sub>1</sub>&nbsp;+ *a*<sub>2</sub>&nbsp;+ …&nbsp;+ *a*<sub>*n*</sub>. | <math>\sum_{k=1}^{4}{k^2}</math> = 1<sup>2</sup>&nbsp;+ 2<sup>2</sup>&nbsp;+ 3<sup>2</sup>&nbsp;+ 4<sup>2</sup>&nbsp;
::= 1&nbsp;+ 4&nbsp;+ 9&nbsp;+ 16&nbsp;= 30 |  |  |
| align=sum over … from … to … of |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>∏</div> | product | <math>\prod_{k=1}^na_k</math> means *a*<sub>1</sub>*a*<sub>2</sub>···*a*<sub>*n*</sub>. | <math>\prod_{k=1}^4(k+2)</math> = (1+2)(2+2)(3+2)(4+2)
::= 3&nbsp;× 4&nbsp;× 5&nbsp;× 6&nbsp;= 360 |  |  |
| align=product over … from … to … of |  |  |  |  |  |
| align=எண்கணிதம் |  |  |  |  |  |
| கார்ட்டீசியன் பெருக்கற்பலன் | <math>\prod_{i=0}^{n}{Y_i}</math> means the set of all (n+1)-tuples
::(*y*<sub>0</sub>, …, *y*<sub>*n*</sub>). | <math>\prod_{n=1}^{3}{\mathbb{R}} = \mathbb{R}\times\mathbb{R}\times\mathbb{R} = \mathbb{R}^3</math> |  |  |  |
| align=the Cartesian product of; the direct product of |  |  |  |  |  |
| align=set theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | coproduct |  |  |  |  |
| align=coproduct over … from … to … of |  |  |  |  |  |
| align=category theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>′

<sup>•</sup></div> | வகையிடல் | *f*&nbsp;′(*x*) is the derivative of the function *f* at the point *x*, i.e., the சாய்வு of the தொடுகோடு to *f* at *x*.

The dot notation indicates a time derivative. That is <math>\dot{x}(t)=\frac{\partial}{\partial t}x(t)</math>. | If *f*(*x*)&nbsp;:=&nbsp;*x*<sup>2</sup>, then *f*&nbsp;′(*x*)&nbsp;=&nbsp;2*x* |  |  |
| align=… prime

derivative of |  |  |  |  |  |
| align=நுண்கணிதம் |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>∫</div> | indefinite integral or antiderivative | ∫&nbsp;*f*(*x*)&nbsp;d*x* means a function whose derivative is *f*. | ∫*x*<sup>2</sup>&nbsp;d*x*&nbsp;= *x*<sup>3</sup>/3 + C |  |  |
| align=indefinite integral of

the antiderivative of |  |  |  |  |  |
| align=நுண்கணிதம் |  |  |  |  |  |
| தொகையீடு | ∫<sub>*a*</sub><sup>*b*</sup>&nbsp;*f*(*x*)&nbsp;d*x* means the signed பரப்பளவு between the *x*-axis and the graph of the function *f* between *x*&nbsp;= *a* and *x*&nbsp;= *b*. | ∫<sub>0</sub><sup>*b*</sup>&nbsp;x<sup>2 </sup>&nbsp;d*x*&nbsp;= *b*<sup>3</sup>/3; |  |  |  |
| align=integral from … to … of … with respect to |  |  |  |  |  |
| align=நுண்கணிதம் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>∮</div> | contour integral or closed line integral | Similar to the integral, but used to denote a single integration over a closed curve or loop. It is sometimes used in physics texts involving equations regarding Gauss's Law, and while these formulas involve a closed surface integral, the representations describe only the first integration of the volume over the enclosing surface. Instances where the latter requires simultaneous double integration, the symbol would be more appropriate. A third related symbol is the closed volume integral, denoted by the symbol .
The contour integral can also frequently be found with a subscript capital letter *C*, ∮<sub>*C*</sub>, denoting that a closed loop integral is, in fact, around a contour *C*, or sometimes dually appropriately, a circle *C*. In representations of Gauss's Law, a subscript capital *S*, ∮<sub>*S*</sub>, is used to denote that the integration is over a closed surface. |  |  |  |
| align=contour integral of |  |  |  |  |  |
| align=நுண்கணிதம் |  |  |  |  |  |
| rowspan=9 bgcolor=#d0f0d0 align=<div>∇</div> | gradient | ∇*f* (x<sub>1</sub>, …, x<sub>*n*</sub>) is the vector of partial derivatives (*∂f* / *∂x*<sub>1</sub>, …, *∂f* / *∂x*<sub>*n*</sub>). | If *f* (*x*,*y*,*z*) := 3*xy* + *z*², then ∇*f*&nbsp;=&nbsp;(3*y*, 3*x*, 2*z*) |  |  |
| align=டெல் இயக்கி, nabla, gradient of |  |  |  |  |  |
| align=vector calculus |  |  |  |  |  |
| விரிதல் (திசையன் நுண்கணிதம்) | <math> \nabla \cdot \vec v = {\partial v_x \over \partial x} + {\partial v_y \over \partial y} + {\partial v_z \over \partial z} </math> | If <math> \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} </math>, then <math> \nabla \cdot \vec v = 3y + 2yz </math>. |  |  |  |
| align=del dot, divergence of |  |  |  |  |  |
| align=vector calculus |  |  |  |  |  |
| curl | <math> \nabla \times \vec v = \( {\partial v_z \over \partial y} - {\partial v_y \over \partial z} \) \mathbf{i} + \( {\partial v_x \over \partial z} - {\partial v_z \over \partial x} \) \mathbf{j} + \( {\partial v_y \over \partial x} - {\partial v_x \over \partial y} \) \mathbf{k} </math> | If <math> \vec v := 3xy\mathbf{i}+y^2 z\mathbf{j}+5\mathbf{k} </math>, then <math> \nabla\times\vec v = -y^2\mathbf{i} - 3x\mathbf{k} </math>. |  |  |  |
| align=curl of |  |  |  |  |  |
| align=vector calculus |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>∂</div> | partial differential | With *f* (x<sub>1</sub>, …, x<sub>*n*</sub>), ∂f/∂x<sub>i</sub> is the derivative of *f* with respect to x<sub>i</sub>, with all other variables kept constant. | If *f*(x,y) := x<sup>2</sup>y, then ∂*f*/∂x = 2xy |  |  |
| align=partial, d |  |  |  |  |  |
| align=நுண்கணிதம் |  |  |  |  |  |
| boundary | ∂*M* means the boundary of *M* | ∂{x : x ≤ 2} = {x : x = 2} |  |  |  |
| align=boundary of |  |  |  |  |  |
| align=இடவியல் |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>&perp;</div> | செங்குத்து | *x*&nbsp;&perp;&nbsp;*y* means *x* is perpendicular to *y*; or more generally *x* is orthogonal to *y*. | If *l*&nbsp;&perp;&nbsp;*m* and *m*&nbsp;&perp;&nbsp;*n* then *l*&nbsp;&nbsp;*n*. |  |  |
| align=is perpendicular to |  |  |  |  |  |
| align=வடிவவியல் |  |  |  |  |  |
| bottom element | *x* = &perp; means *x* is the smallest element. | &forall;*x*&nbsp;: *x*&nbsp;∧&nbsp;&perp;&nbsp;= &perp; |  |  |  |
| align=the bottom element |  |  |  |  |  |
| align=lattice theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | சமாந்தரம் | *x*&nbsp;&nbsp;*y* means *x* is parallel to *y*. | If *l*&nbsp;&nbsp;*m* and *m*&nbsp;&perp;&nbsp;*n* then *l*&nbsp;&perp;&nbsp;*n*. |  |  |
| align=is parallel to |  |  |  |  |  |
| align=வடிவவியல் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align= <div></div> | entailment | *A*&nbsp;&nbsp;*B* means the sentence *A* entails the sentence *B*, that is in every model in which *A* is true, *B* is also true. | *A*&nbsp;&nbsp;*A*&nbsp;∨&nbsp;¬*A* |  |  |
| align=entails |  |  |  |  |  |
| align= model theory |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align= <div></div> | inference | *x*&nbsp;&nbsp;*y* means *y* is derived from *x*. | *A*&nbsp;→&nbsp;*B*&nbsp; ¬*B*&nbsp;→&nbsp;¬*A* |  |  |
| align=infers or is derived from |  |  |  |  |  |
| align=propositional logic, predicate logic |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align= <div></div> | இயல்நிலை உட்குலம் | *N*&nbsp;&nbsp;*G* means that *N* is a normal subgroup of group *G*. | *Z*(*G*)&nbsp;&nbsp;*G* |  |  |
| align=is a normal subgroup of |  |  |  |  |  |
| align=குலக் கோட்பாடு |  |  |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>/</div> | ஈவு குலம் | *G*&nbsp;/&nbsp;*H* means the quotient of group *G* modulo its subgroup *H*. | {0, *a*, 2*a*, *b*, *b*+*a*, *b*+2*a*}&nbsp;/&nbsp;{0, *b*}&nbsp;= , {*a*, *b*+*a*}, |  |  |
| align= mod |  |  |  |  |  |
| align= குலக் கோட்பாடு |  |  |  |  |  |
| quotient set | *A*/~ means the set of all ~ சமானப் பகுதிes in *A*. | If we define ~ by x&nbsp;~&nbsp;y ⇔ x&nbsp;−&nbsp;y&nbsp;∈ , then
/~&nbsp;= |  |  |  |
| rowspan=6 bgcolor=#d0f0d0 align=<div>≈</div> | approximately equal | *x*&nbsp;≈&nbsp;*y* means *x* is approximately equal to *y*. | π&nbsp;≈&nbsp;3.14159 |  |  |
| align=is approximately equal to |  |  |  |  |  |
| align=everywhere |  |  |  |  |  |
| isomorphism | *G*&nbsp;≈&nbsp;*H* means that group *G* is isomorphic to group *H*. | *Q*&nbsp;/&nbsp;{1,&nbsp;−1}&nbsp;≈ *V*,
where *Q* is the quaternion group and *V* is the கிளைன் நான்குறுப்புக்குலம். |  |  |  |
| is isomorphic to |  |  |  |  |  |
| align= குலக் கோட்பாடு |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>~</div> | same order of magnitude | *m*&nbsp;~&nbsp;*n* means the quantities *m* and *n* have the same order of magnitude, or general size.

(*Note that* ~ *is used for an approximation that is poor, otherwise use * ≈&nbsp;.) | 2&nbsp;~&nbsp;5

8&nbsp;×&nbsp;9&nbsp;~ 100

but π<sup>2</sup> ≈ 10 |  |  |
| align=roughly similar

poorly approximates |  |  |  |  |  |
| align=அண்ணளவாக்கக் கோட்பாடு |  |  |  |  |  |
| )

< , >

·

:</div> | inner product | 〈*x*,*y*〉 means the inner product of *x* and *y* as defined in an inner product space.

For spatial vectors, the புள்ளிப் பெருக்கல் notation, *x*·*y* is common.

For matricies, the colon notation may be used. | The standard inner product between two vectors *x*&nbsp;=&nbsp;(2,&nbsp;3) and *y*&nbsp;=&nbsp;(−1,&nbsp;5) is:
〈x,&nbsp;y〉&nbsp;=&nbsp;2&nbsp;×&nbsp;−1&nbsp;+&nbsp;3&nbsp;×&nbsp;5&nbsp;= 13

<math>A:B = \sum_{i,j} A_{ij}B_{ij}</math> |  |  |
| align=inner product of |  |  |  |  |  |
| align=நேரியல் இயற்கணிதம் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div></div> | tensor product | *V*&nbsp;&nbsp;*U* means the tensor product of *V* and *U*. | {1, 2, 3, 4}&nbsp;&nbsp;{1, 1, 2}&nbsp;=
, {1, 2, 3, 4}, |  |  |
| align= tensor product of |  |  |  |  |  |
| align= நேரியல் இயற்கணிதம் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div>*</div> | convolution | *f*&nbsp;*&nbsp;*g* means the convolution of *f* and *g*. | <math>(f * g )(t) = \int f(\tau) g(t - \tau)\, d\tau</math> |  |  |
| align= convolution, convoluted with |  |  |  |  |  |
| align= functional analysis |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div><math>\bar{x}</math>
</div> | கூட்டுச்சராசரி | <math>\bar{x}</math> (often read as "x bar") is the கூட்டுச்சராசரி (average value of <math>x_i</math>). | <math>x = \{1,2,3,4,5\}; \bar{x} = 3</math>. |  |  |
| align=overbar, … bar |  |  |  |  |  |
| align=புள்ளியியல் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div><math> \overline{z} </math> | complex conjugate | <math> \overline{z} </math> is the complex conjugate of *z*. | <math> \overline{3+4i} = 3-4i </math> |  |  |
| align=conjugate |  |  |  |  |  |
| align=சிக்கலெண் |  |  |  |  |  |
| rowspan=3 bgcolor=#d0f0d0 align=<div><math>\triangleq</math></div> | delta equal to | <math>\triangleq</math> means equal by definition. When <math>\triangleq</math> is used, equality is not true generally, but rather equality is true under certain assumptions that are taken in context. Some writers prefer ≡. | x_{\pi_i})</math>. |  |  |
| align=equal by definition |  |  |  |  |  |
| align=everywhere | } |  |  |  |  |