# Join and meet: Exercises

Try these exercises to test your knowledge of joins and meets.

## Tangled up

Determine whether or not the following joins and meets exist in the poset depicted by the above Hasse diagram.

\(c \vee b\)

*not*exist, because \(i\) and \(j\) are incomparable upper bounds of \(\{ c, b \}\), and no smaller upper bounds of \(\{ c, b \}\) exist.

\(g \vee e\)

\(j \wedge f\)

*not*exist, because \(d\) and \(b\) are incomparable lower bounds of \(\{ j, f \}\), and no larger lower bounds of \(\{ j, f \}\) exist.

\(\bigvee \{a,b,c,d,e,f,g,h,i,j,k,l\}\)

## Join fu

Let \(P\) 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 \((\bigvee S) \vee p = \bigvee (S \cup \{p\})\).

\(q \in \{\bigvee S, p\}^U \iff\)

for all \(s \in S, q \geq \bigvee S \geq s\), and \(q \geq p \iff\)

\(q \in (S \cup \{p\})^U\).

If 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. <div><div>

## Meet fu

Let \(P\) 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 \((\bigwedge S) \wedge p = \bigwedge(S \cup \{p\})\).

## Quite a big join

In the poset \(\langle \mathbb N, | \rangle\) discussed in Poset: Examples, does \(\bigvee \mathbb N\) exist? If so, what is it?

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