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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\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\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',
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