{ localUrl: '../page/relation_mathematics.html', arbitalUrl: 'https://arbital.com/p/relation_mathematics', rawJsonUrl: '../raw/3nt.json', likeableId: '2527', likeableType: 'page', myLikeValue: '0', likeCount: '2', dislikeCount: '0', likeScore: '2', individualLikes: [ 'EricBruylant', 'BrettHoutz' ], pageId: 'relation_mathematics', edit: '11', editSummary: '', prevEdit: '10', currentEdit: '11', wasPublished: 'true', type: 'wiki', title: 'Relation', clickbait: '', textLength: '1811', alias: 'relation_mathematics', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'DylanHendrickson', editCreatedAt: '2016-07-07 17:11:14', pageCreatorId: 'KevinClancy', pageCreatedAt: '2016-05-17 00:48:51', seeDomainId: '0', editDomainId: 'AlexeiAndreev', submitToDomainId: '0', isAutosave: 'false', isSnapshot: 'false', isLiveEdit: 'true', isMinorEdit: 'false', indirectTeacher: 'false', todoCount: '1', isEditorComment: 'false', isApprovedComment: 'true', isResolved: 'false', snapshotText: '', anchorContext: '', anchorText: '', anchorOffset: '0', mergedInto: '', isDeleted: 'false', viewCount: '63', text: '%%comment:\nI do not want to be shortened. The motivation for this is that I would prefer that someone has the ability to learn everything they need to know about relations just by reading the popup summary.\n%%\n\n[summary: A **relation** is a [3jz set] of [tuple_mathematics tuples], all of which have the same [tuple_arity arity]. The inclusion of a tuple in a relation indicates that the components of the tuple are related. A set of $n$-tuples is called an $n$*-ary relation*. Sets of pairs are called binary relations, sets of triples are called ternary relations, etc.\n\nExamples of binary relations include the equality relation on natural numbers $\\{ (0,0), (1,1), (2,2), ... \\}$ and the predecessor relation $\\{ (0,1), (1,2), (2,3), ... \\}$. When a symbol is used to denote a specific binary relation ($R$ is commonly used for this purpose), that symbol can be used with infix notation to denote set membership: $xRy$ means that the pair $(x,y)$ is an element of the set $R$.]\n\nA **relation** is a [3jz set] of [tuple_mathematics tuples], all of which have the same [todo: generalize the function_arity page to include general arity][tuple_arity arity]. The inclusion of a tuple in a relation indicates that the components of the tuple are related. A set of $n$-tuples is called an $n$*-ary relation*. Sets of pairs are called binary relations, sets of triples are called ternary relations, etc.\n\nExamples of binary relations include the equality relation on natural numbers $\\{ (0,0), (1,1), (2,2), ... \\}$ and the predecessor relation $\\{ (0,1), (1,2), (2,3), ... \\}$. When a symbol is used to denote a specific binary relation ($R$ is commonly used for this purpose), that symbol can be used with infix notation to denote set membership: $xRy$ means that the pair $(x,y)$ is an element of the set $R$.', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: 'null', muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'true', proposalEditNum: '0', permissions: { edit: { has: 'false', reason: 'You don't have domain permission to edit this page' }, proposeEdit: { has: 'true', reason: '' }, delete: { has: 'false', reason: 'You don't have domain permission to delete this page' }, comment: { has: 'false', reason: 'You can't comment in this domain because you are not a member' }, proposeComment: { has: 'true', reason: '' } }, summaries: {}, creatorIds: [ 'KevinClancy', 'DylanHendrickson', 'NateSoares', 'JoeZeng' ], childIds: [ 'equivalence_relation', 'order_relation', 'transitive_relation', 'reflexive_relation', 'antisymmetric_relation' ], parentIds: [ 'math' ], commentIds: [], questionIds: [], tagIds: [ 'formal_definition_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [ { id: '3236', parentId: 'set_mathematics', childId: 'relation_mathematics', type: 'requirement', creatorId: 'AlexeiAndreev', createdAt: '2016-06-17 21:58:56', level: '1', isStrong: 'false', everPublished: 'true' } ], subjects: [], lenses: [], lensParentId: '', pathPages: [], learnMoreTaughtMap: {}, learnMoreCoveredMap: {}, learnMoreRequiredMap: {}, editHistory: {}, domainSubmissions: {}, answers: [], answerCount: '0', commentCount: '0', newCommentCount: '0', linkedMarkCount: '0', changeLogs: [ { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '17850', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '0', type: 'newChild', createdAt: '2016-07-31 17:53:22', auxPageId: 'antisymmetric_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16806', pageId: 'relation_mathematics', userId: 'AlexeiAndreev', edit: '0', type: 'newChild', createdAt: '2016-07-15 20:17:42', auxPageId: 'reflexive_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16804', pageId: 'relation_mathematics', userId: 'AlexeiAndreev', edit: '0', type: 'deleteChild', createdAt: '2016-07-15 20:17:23', auxPageId: 'reflexive_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16801', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'deleteUsedAsTag', createdAt: '2016-07-15 20:00:17', auxPageId: 'reflexive_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16799', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'newChild', createdAt: '2016-07-15 20:00:16', auxPageId: 'reflexive_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16007', pageId: 'relation_mathematics', userId: 'DylanHendrickson', edit: '0', type: 'newChild', createdAt: '2016-07-07 17:20:32', auxPageId: 'transitive_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16002', pageId: 'relation_mathematics', userId: 'DylanHendrickson', edit: '11', type: 'newEdit', createdAt: '2016-07-07 17:11:14', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '16001', pageId: 'relation_mathematics', userId: 'DylanHendrickson', edit: '10', type: 'newEdit', createdAt: '2016-07-07 17:10:58', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15698', pageId: 'relation_mathematics', userId: 'JoeZeng', edit: '0', type: 'newChild', createdAt: '2016-07-06 15:24:37', auxPageId: 'order_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15482', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '9', type: 'newEdit', createdAt: '2016-07-05 22:25:30', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15452', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'newChild', createdAt: '2016-07-05 21:52:51', auxPageId: 'equivalence_relation', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '15426', pageId: 'relation_mathematics', userId: 'JoeZeng', edit: '7', type: 'newEdit', createdAt: '2016-07-05 20:30:02', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13766', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'newAlias', createdAt: '2016-06-17 23:21:21', auxPageId: '', oldSettingsValue: '3nt', newSettingsValue: 'relation_mathematics' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13765', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'newTag', createdAt: '2016-06-17 23:21:14', auxPageId: 'formal_definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '13763', pageId: 'relation_mathematics', userId: 'EricBruylant', edit: '0', type: 'deleteTag', createdAt: '2016-06-17 23:21:08', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10775', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '6', type: 'newRequiredBy', createdAt: '2016-05-21 22:00:13', auxPageId: 'poset', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10556', pageId: 'relation_mathematics', userId: 'NateSoares', edit: '6', type: 'newEdit', createdAt: '2016-05-17 06:30:48', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10554', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '5', type: 'newEdit', createdAt: '2016-05-17 02:33:35', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10553', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '4', type: 'newEdit', createdAt: '2016-05-17 02:32:49', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10552', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '3', type: 'newEdit', createdAt: '2016-05-17 02:31:35', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10551', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '2', type: 'newEdit', createdAt: '2016-05-17 02:29:59', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10531', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '1', type: 'newEdit', createdAt: '2016-05-17 00:48:51', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10527', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '1', type: 'newRequirement', createdAt: '2016-05-17 00:46:42', auxPageId: 'set_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10526', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '1', type: 'newTag', createdAt: '2016-05-17 00:46:04', auxPageId: 'definition_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '10525', pageId: 'relation_mathematics', userId: 'KevinClancy', edit: '1', type: 'newParent', createdAt: '2016-05-17 00:45:58', auxPageId: 'math', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'true', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: {} }