Exercise · ZW-HASSE-DIAGRAM-010 · ★★★ Advanced
What is the largest set of divisors of \(30\) that are pairwise incomparable (an antichain) under divisibility?
Choose one option.
Show answer
- B. 3
Explanation
For example \(\{2,3,5\}\) or \(\{6,10,15\}\); with four elements, two would always be comparable.
Review the concept: Hasse diagram
A Hasse diagram draws a partially ordered set \((S,\preceq)\) with one point per element and an edge only between consecutive elements, placing the smaller one lower.
\[a\ \text{covers}\ b\iff b\prec a\ \wedge\ \nexists\,c:\ b\prec c\prec a\]
More practice problems: hasse diagram
- ZW-HASSE-DIAGRAM-001 In the Hasse diagram of the divisors of 12 under divisibility, how many edges are there?
- ZW-HASSE-DIAGRAM-002 In the Hasse diagram of the divisors of 30, how many elements directly cover 1?
- ZW-HASSE-DIAGRAM-003 In the divisors of 12 under divisibility, which pair is an edge of the Hasse diagram?
- ZW-HASSE-DIAGRAM-004 In \2,3,4,6,8,12\ under divisibility, how many maximal elements does the diagram have (nothing abov…
- ZW-HASSE-DIAGRAM-005 Why does a Hasse diagram draw no loops and no edges that follow by transitivity?
- ZW-HASSE-DIAGRAM-006 How many edges does the Hasse diagram of the divisors of 30 under divisibility have?
- ZW-HASSE-DIAGRAM-007 How many edges does the Hasse diagram of \1,2,4,8,16,32\ under divisibility have?
- ZW-HASSE-DIAGRAM-008 In the Hasse diagram of the divisors of 12, which pair of elements is comparable (an upward path jo…