Operation
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Revision 2507
8/28/2026, 4:49:19 PM · Catalpa
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Revision 2505
8/28/2026, 4:46:56 PM · Catalpa
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##### Intuition2
An operation is a process aiming at obtaining a result through some manipulations of mathematical objects or values: addition, substraction, multiplication and division are the most commonly known operations.3
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##### Formal definition1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a [[Function|function]] defined on some cartesian product $X\times\dots\times X$ and which [[Image of a map|image]] is included in $X$.2
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##### Remarks and examples4
- If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.8
- If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation - like addition, substraction, multiplication...5
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- Addition, substraction, multiplication and division are common examples of binary operations.7
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- The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is an operation on the set of integers. 9
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- The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 836
8/2/2026, 9:35:39 AM · Ancient Tree
Concept edited
Compare with revision 8359 changed lines
1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a [[Function|function]] defined on some cartesian product $X\times\dots\times X$ and which [[Image of a map|image]] is included in $X$.2
3
**Remarque:**3
##### Remarks and examples4
- If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.4
5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.6
- Addition, substraction, multiplication and division are common examples of binary operations.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
- The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is an operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.10
- The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 835
8/2/2026, 9:32:31 AM · Ancient Tree
Concept reviewed
This older revision predates detailed metadata tracking.
Revision 834
8/2/2026, 9:32:24 AM · Ancient Tree
Concept edited
Compare with revision 8332 changed lines
1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a [[Function|function]] defined on some cartesian product $X\times\dots\times X$ and which [[Image of a map|image]] is included in $X$.2
3
**Remarque:**4
5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 833
8/2/2026, 9:31:41 AM · Ancient Tree
Concept edited
Compare with revision 8322 changed lines
1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.2
3
**Remarque:**4
5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^4\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^3\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 832
8/2/2026, 7:45:14 AM · Sequoia
Concept edited
Compare with revision 8154 changed lines
1
Let $E$ be a set. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.1
Let $E$ be a [[Set|set]]. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.2
3
**Remarque:**4
5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation, or a law.5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^4\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 815
8/1/2026, 5:43:17 PM · Sequoia
Concept edited
Compare with revision 8112 changed lines
1
Let $E$ be a set. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.2
3
**Remarque:**4
5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then one talks about binary operation.5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then $*$ is called a binary operation, or a law.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^4\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 811
8/1/2026, 5:41:15 PM · Sequoia
Concept edited
This older revision predates detailed metadata tracking.
Revision 810
8/1/2026, 5:40:49 PM · Sequoia
Concept edited
Compare with revision 8092 changed lines
1
Let $E$ be a set. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.2
3
**Remarque:**4
5
$\bullet$ Si le produit cartésien est un produit de $X$ et lui-même, soit $*\colon X\times X\longrightarrow X$, on parle de binary operation.5
$\bullet$ If the cartesian product is a product of $X$ and itself, hence $*\colon X\times X\longrightarrow X$, then one talks about binary operation.6
7
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^4\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers. 8
9
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.Revision 809
8/1/2026, 5:39:45 PM · Sequoia
Concept created
Let $E$ be a set. An operation $*$ on $X$ is a function defined on some cartesian product $X\times\dots\times X$ and which image is included in $X$.
**Remarque:**
$\bullet$ Si le produit cartésien est un produit de $X$ et lui-même, soit $*\colon X\times X\longrightarrow X$, on parle de binary operation.
$\bullet$ The function $\displaystyle *\colon (a,b,c)\in\Z^4\longmapsto a+b-c\in \Z$ is a binary operation on the set of integers.
$\bullet$ The function $\displaystyle *\colon (a,b,c,d)\in\R^4\longmapsto \frac{2^a+b}{c^2+d^2+1}\in \R$ is an operation on the set of real numbers.