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text: 'Try these exercises to test your knowledge of joins and meets.\n\n\nTangled up \n--------------------\n\n\n\nDetermine whether or not the following joins and meets exist in the poset depicted by the above Hasse diagram. \n\n$c \\vee b$\n\n%%hidden(Show solution):\nThis join does *not* exist, because $i$ and $j$ are incomparable upper bounds of $\\{ c, b \\}$, and no smaller upper bounds of $\\{ c, b \\}$ exist.\n%%\n\n$g \\vee e$\n\n%%hidden(Show solution):\nThis join exists. $g \\vee e = j$.\n%%\n\n$j \\wedge f$ \n\n%%hidden(Show solution):\nThis meet does *not* exist, because $d$ and $b$ are incomparable lower bounds of $\\{ j, f \\}$, and no larger lower bounds of $\\{ j, f \\}$ exist.\n%%\n\n$\\bigvee \\{a,b,c,d,e,f,g,h,i,j,k,l\\}$\n\n%%hidden(Show solution):\nThis join exists. It is $l$.\n%%\n\nJoin fu\n------------\n\nLet $P$ be a poset, $S \\subseteq P$, and $p \\in P$. Prove that if both $\\bigvee S$ and $(\\bigvee S) \\vee p$ exist then $\\bigvee (S \\cup \\{p\\})$ exists as well, and $(\\bigvee S) \\vee p = \\bigvee (S \\cup \\{p\\})$.\n\n%%hidden(Show solution):\nFor any $X \\subset P$, let $X^U$ denote the set of upper bounds of $X$. The above proposition follows from the fact that $\\{\\bigvee S, p\\}^U = (S \\cup p)^U$, which is apparent from the following chain of bi-implications:\n\n$q \\in \\{\\bigvee S, p\\}^U \\iff$ \n\nfor all $s \\in S, q \\geq \\bigvee S \\geq s$, and $q \\geq p \\iff$\n\n$q \\in (S \\cup \\{p\\})^U$.\n\nIf two subsets of a poset have the same set of upper bounds, then either they both lack a least upper bound, or both have the same least upper bound.\n%%\n\nMeet fu\n------------\n\nLet $P$ be a poset, $S \\subseteq P$, and $p \\in P$. Prove that if both $\\bigwedge S$ and $(\\bigwedge S) \\wedge p$ exist then $\\bigwedge (S \\cup \\{p\\})$ exists as well, and $(\\bigwedge S) \\wedge p = \\bigwedge(S \\cup \\{p\\})$.\n\n%%hidden(Show solution):\nNote that the proposition we are trying to prove here is the dual of the one stated in join fu. Thanks to the duality principle, this theorem therefore comes for free with our solution to join fu.\n%%\n\nQuite a big join\n--------------------------\n\nIn the poset $\\langle \\mathbb N, | \\rangle$ discussed in [43s], does $\\bigvee \\mathbb N$ exist? If so, what is it?\n\n',
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