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"wiki", "title": "Function", "clickbait": "", "textLength": 2093, "alias": "function", "externalUrl": "", "sortChildrenBy": "likes", "hasVote": false, "voteType": "", "votesAnonymous": false, "editCreatorId": "5", "editCreatedAt": "2016-06-08 19:21:49", "pageCreatorId": "32", "pageCreatedAt": "2016-05-12 18:32:12", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 6, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 182, "text": "Intuitively, a function $f$ is a procedure (or machine) that takes an input and performs some operations to produce an output. For example, the function \"+\" takes a pair of numbers as input and produces their sum as output: on input (3, 6) it produces 9 as output, on input (2, 18) it produces 20 as output, and so on.\n\nFormally, in mathematics, a function $f$ is a relationship between a [3jz set] $X$ of inputs and a set $Y$ of outputs, which relates each input to exactly one output. For example, $-$ is a function that relates the pair $(4, 3)$ to $1,$ and $(19, 2)$ to $17,$ and so on. In this case, the input set is all possible pairs of [number numbers], and the output set is numbers. We write [3vl $f : X \\to Y$] (and say \"$f$ has the [type_mathematics type] $X$ to $Y$\") to denote that $f$ is some function that relates inputs from the set $X$ to outputs from the set $Y$. 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"2016-05-14 16:23:46", "seeDomainId": "0", "editDomainId": "1", "submitToDomainId": "0", "isAutosave": false, "isSnapshot": false, "isLiveEdit": true, "isMinorEdit": false, "indirectTeacher": false, "todoCount": 0, "isEditorComment": false, "isApprovedComment": true, "isResolved": false, "snapshotText": "", "anchorContext": "", "anchorText": "", "anchorOffset": 0, "mergedInto": "", "isDeleted": false, "viewCount": 92, "text": "[summary:Many functions can be visualized as physical mechanisms of wheels and gears, that take their inputs on conveyor belts, manipulate them (using mechanical sensors and tools), and produce an output which is placed on an outgoing conveyor belt.\n\n![A binary function](http://i.imgur.com/lKxAXcK.png)\n\nFor example, we can visualize the function $+$ as a function that takes two stacks of poker chips in as input (one on the left conveyor belt and one on the right conveyor belt) and produces its output by dumping both stacks onto the output belt.]\n\nMany functions can be visualized as physical mechanisms of wheels and gears, that take their inputs on conveyor belts, manipulate them (using mechanical sensors and tools), and produce an output which is placed on an outgoing conveyor belt.\n\n![A binary function](http://i.imgur.com/lKxAXcK.png)\n\nFor example, we can visualize the function $+$ as a function that takes two stacks of poker chips in as input (one on the left conveyor belt and one on the right conveyor belt) and produces its output by dumping both stacks onto the output belt. We can visualize the function $\\times$ as a function that also gets a stack of poker chips on each input belt, and which puts a copy of the left stack onto the output tape for each poker chip in the right stack.\n\nAll [computable_function computable] functions can (in principle) be implemented by a physical system of wheels and gears (or circuits and wires, etc.), and it is conjectured that the computable functions are the _only_ functions that can be implemented by physical systems. 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I suggest you merge the two and do what you want with them. 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