{ localUrl: '../page/set_product.html', arbitalUrl: 'https://arbital.com/p/set_product', rawJsonUrl: '../raw/5zs.json', likeableId: '0', likeableType: 'page', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], pageId: 'set_product', edit: '2', editSummary: '', prevEdit: '1', currentEdit: '2', wasPublished: 'true', type: 'wiki', title: 'Set product', clickbait: 'A fundamental way of combining sets is to take their product, making a set that contains all tuples of elements from the originals.', textLength: '1436', alias: 'set_product', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'PatrickStevens', editCreatedAt: '2016-08-26 12:46:41', pageCreatorId: 'PatrickStevens', pageCreatedAt: '2016-08-25 07:47:47', seeDomainId: '0', editDomainId: 'AlexeiAndreev', 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: '1881', text: '[summary: The product of two [3jz sets] $A$ and $B$ is just the collection of ordered pairs $(a,b)$ where $a$ is in $A$ and $b$ is in $B$. The reason we call it the "product" can be seen if you consider the set-product of $\\{1,2,\\dots,n \\}$ and $\\{1,2,\\dots, m \\}$: it consists of ordered pairs $(a, b)$ where $1 \\leq a \\leq n$ and $1 \\leq b \\leq m$, but if we interpret these as integer coordinates in the plane, we obtain just an $n \\times m$ rectangle.]\n\n[summary(Technical): The product of [3jz sets] $Y_x$ indexed by the set $X$ is denoted $\\prod_{x \\in X} Y_x$, and it consists of all $X$-length ordered tuples of elements. For example, if $X = \\{1,2\\}$, and $Y_1 = \\{a,b\\}, Y_2 = \\{b,c\\}$, then $$\\prod_{x \\in X} Y_x = Y_1 \\times Y_2 = \\{(a,b), (a,c), (b,b), (b,c)\\}$$\nIf $X = \\mathbb{Z}$ and $Y_n = \\{ n \\}$, then $$\\prod_{x \\in X} Y_x = \\{(\\dots, -2, -1, 1, 0, 1, 2, \\dots)\\}]\n\n[todo: define the product as tuples]\n\n[todo: several examples, including R^n being the product over $\\{1,2, \\dots, n\\}$; this introduces associativity of the product which is covered later]\n\n[todo: product is associative up to isomorphism, though not literally]\n\n[todo: cardinality of the product, noting that in the finite case it collapses to just the usual definition of the product of natural numbers]\n\n[todo: as an aside, define the product formally in ZF]\n\n[todo: link to universal property, mentioning it is a product in the category of sets]', 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: 'false', proposalEditNum: '3', 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: [ 'PatrickStevens', 'AntonChaynikov' ], childIds: [], parentIds: [ 'set_mathematics' ], commentIds: [], questionIds: [], tagIds: [ 'needs_summary_meta_tag', 'stub_meta_tag' ], relatedIds: [], markIds: [], explanations: [], learnMore: [], requirements: [], 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: '23232', pageId: 'set_product', userId: 'AntonChaynikov', edit: '3', type: 'newEditProposal', createdAt: '2019-11-18 10:33:02', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3447', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '19163', pageId: 'set_product', userId: 'PatrickStevens', edit: '2', type: 'newEdit', createdAt: '2016-08-26 12:46:41', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19128', pageId: 'set_product', userId: 'PatrickStevens', edit: '0', type: 'newTag', createdAt: '2016-08-25 07:47:50', auxPageId: 'stub_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19129', pageId: 'set_product', userId: 'PatrickStevens', edit: '0', type: 'newTag', createdAt: '2016-08-25 07:47:50', auxPageId: 'needs_summary_meta_tag', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '0', likeableType: 'changeLog', myLikeValue: '0', likeCount: '0', dislikeCount: '0', likeScore: '0', individualLikes: [], id: '19127', pageId: 'set_product', userId: 'PatrickStevens', edit: '0', type: 'newParent', createdAt: '2016-08-25 07:47:49', auxPageId: 'set_mathematics', oldSettingsValue: '', newSettingsValue: '' }, { likeableId: '3435', likeableType: 'changeLog', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '19125', pageId: 'set_product', userId: 'PatrickStevens', edit: '1', type: 'newEdit', createdAt: '2016-08-25 07:47:47', auxPageId: '', oldSettingsValue: '', newSettingsValue: '' } ], feedSubmissions: [], searchStrings: {}, hasChildren: 'false', hasParents: 'true', redAliases: {}, improvementTagIds: [], nonMetaTagIds: [], todos: [], slowDownMap: 'null', speedUpMap: 'null', arcPageIds: 'null', contentRequests: { improveStub: { likeableId: '3585', likeableType: 'contentRequest', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [], id: '108', pageId: 'set_product', requestType: 'improveStub', createdAt: '2016-10-08 14:50:22' } } }