Consider the set \(X=\{a, b, c, d, e\}\) under partial ordering \(R=\{(a,a),…
2017
Consider the set \(X=\{a, b, c, d, e\}\) under partial ordering
\(R=\{(a,a), (a, b), (a, c), (a, d), (a, e), (b, b), (b, c), (b, e), (c, c), (c, e), (d, d), (d, e), (e, e) \}\)
The Hasse diagram of the partial order \((X, R)\) is shown below.
The minimum number of ordered pairs that need to be added to \(R\) to make \((X, R)\) a lattice is ______
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Correct answer: 0
Answer: 0 (No ordered pairs need to be added — the relation is already a lattice.)
Key observations: a is the minimum (bottom) element and e is the maximum (top) element of the poset.
If two elements are comparable (one ≤ the other), their meet is the smaller and their join is the larger; these are already in X.
For the incomparable pairs, check common bounds explicitly: for b and d, the common upper bounds are {e} so join(b,d)=e, and the common lower bounds are {a} so meet(b,d)=a. The same reasoning gives join(c,d)=e and meet(c,d)=a.
Since every pair of elements has a well-defined meet and join in X, (X,R) is already a lattice.
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