{ localUrl: '../page/join_examples.html', arbitalUrl: 'https://arbital.com/p/join_examples', rawJsonUrl: '../raw/3v4.json', likeableId: '2771', likeableType: 'page', myLikeValue: '0', likeCount: '1', dislikeCount: '0', likeScore: '1', individualLikes: [ 'EricBruylant' ], pageId: 'join_examples', edit: '19', editSummary: '', prevEdit: '18', currentEdit: '19', wasPublished: 'true', type: 'wiki', title: 'Join and meet: Examples', clickbait: '', textLength: '2833', alias: 'join_examples', externalUrl: '', sortChildrenBy: 'likes', hasVote: 'false', voteType: '', votesAnonymous: 'false', editCreatorId: 'KevinClancy', editCreatedAt: '2016-06-20 03:57:20', pageCreatorId: 'KevinClancy', pageCreatedAt: '2016-05-27 02:43:38', 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: '117', text: 'A union of sets and the least common multiple of a set of natural numbers can both be viewed as joins. In addition, joins can be useful to the designers of statically typed programming languages.\n\n%%%knows-requisite([3v1]):\nThe real numbers\n------------------------------\n\nConsider the [-3rb] $\\langle \\mathbb{R}, \\leq \\rangle$ of all real numbers ordered by the standard comparison relation. For any non-empty $X \\subseteq \\mathbb{R}$, $\\bigvee X$ exists if and only if $X$ has an upper bound; this fact falls directly out of the definition of the set of real numbers.\n%%%\n\n\nSubtyping\n------------------\n\nStatically typed programming languages often define a poset of types ordered by the subtyping relation; Scala is one such language. Consider the following Scala program.\n\n![A scala program with class inheritance](http://i.imgur.com/IUkzT7A.png)\n\nWhen a programmer defines a class hierarchy in an object-oriented language, they are actually defining a poset of types. The above program defines the simple poset shown in the following Hasse diagram.\n\n![Hasse diagram for the nominal subtyping relation defined by the above program](http://i.imgur.com/dk1EHuA.png)\n\n%%%comment:\ndot source:\n\ndigraph G {\n node [width = 0.1, height = 0.1];\n edge [arrowhead = "none"];\n rankdir = BT;\n Dog -> Animal;\n Cat -> Animal;\n}\n%%%\n\nNow consider the expression *if (b) dog else cat*. If b is true, then it evaluates to a value of type Dog. If b is false, then it evaluates to a value of type Cat. What type, then, should *if (b) dog else cat* have? Its type should be the join of Dog and Cat, which is Animal.\n\nPower sets\n-------------------\n\nLet $X$ be a [-3jz]. Consider the [-3rb] $\\langle \\mathcal{P}(X), \\subseteq \\rangle$, the power set of $X$ ordered by inclusion. In this poset, joins are unions: for all $A \\subseteq \\mathcal{P}(X)$, $\\bigvee A = \\bigcup A$. This can be shown as follows. Let $A \\subseteq \\mathcal{P}(X)$. Then $\\bigcup A$ is an upper bound of $A$ because a union contains each of its constituent sets. Furthermore, $\\bigcup A$ is the *least* upper bound of $A$. For let $Z$ be an upper bound of $A$. Then $x \\in \\bigcup A$ implies $x \\in Y$ for some $Y \\in A$, and since $Y \\subseteq Z$, we have $x \\in Y \\subseteq Z$. Since $x \\in \\bigcup A$ implies $x \\in Z$, we have $\\bigcup A \\subseteq Z$. Hence, $\\bigvee A = \\bigcup A$.\n\nDivisibility\n------------------\n\nConsider the poset $\\langle \\mathbb Z_+, | \\rangle$ of divisibility on the positive integers. In this poset, the upper bounds of an integer are exactly its multiples. Thus, the join of a set of positive integers in $\\langle \\mathbb Z_+, | \\rangle$ is their [least_common_multiple least common multiple]. Dually, the meet of a set of positive integers in $\\langle \\mathbb Z_+, | \\rangle$ is their [greatest_common_divisor greatest common divisor].\n\n\n\n', metaText: '', isTextLoaded: 'true', isSubscribedToDiscussion: 'false', isSubscribedToUser: 'false', isSubscribedAsMaintainer: 'false', discussionSubscriberCount: '1', maintainerCount: '1', userSubscriberCount: '0', lastVisit: '', hasDraft: 'false', votes: [], voteSummary: [ '0', '0', '0', '0', '0', '0', '0', '0', '0', '0' ], muVoteSummary: '0', voteScaling: '0', currentUserVote: '-2', voteCount: '0', lockedVoteType: '', maxEditEver: '0', redLinkCount: '0', lockedBy: '', lockedUntil: '', nextPageId: '', prevPageId: '', usedAsMastery: 'false', 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 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