{ localUrl: '../page/colon_to_notation.html', arbitalUrl: 'https://arbital.com/p/colon_to_notation', rawJsonUrl: '../raw/3vl.json', likeableId: '2560', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricRogstad' ], pageId: 'colon_to_notation', edit: '12', editSummary: '', prevEdit: '11', currentEdit: '12', wasPublished: 'true', type: 'wiki', title: 'Colon-to notation', clickbait: 'Find out what the notation "f : X -> Y" means that everyone keeps using. ', textLength: '821', alias: 'colon_to_notation', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'DylanHendrickson', editCreatedAt: '2016-08-04 16:54:29', pageCreatorId: 'QiaochuYuan', pageCreatedAt: '2016-05-27 18:15:32', seeDomainId: '0', editDomainId: 'AlexeiAndreev', 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: '68', text: 'In mathematics, the notation $f : X \\to Y$ (here, "colon-to notation," because the arrow $\\to$ is written "\\to" in [LaTeX]) means that $f$ is a [3jy function] with [3js domain] $X$ and [3lg codomain] $Y$. It can be read "$f$, a function from $X$ to $Y$." \n\nThis can be thought of as ascribing a function type to the value $f$. The use of a colon to express that a given value has a given type, as is done in type theory, is a generalization of this notation.\n\n# Examples\n\n$f : \\mathbb{R} \\to \\mathbb{R}$ means that $f$ is a function from the real numbers to the real numbers, such as $x \\mapsto x^2$ ([3vm mapsto notation]). \n\n$f : \\mathbb{R} \\times \\mathbb{R} \\to \\mathbb{R}$ means that $f$ is a function from pairs of real numbers to real numbers. The $\\times$ here refers to the [3xb Cartesian product] of [3jz sets]. 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