Final answer:
A Hasse diagram is a graphical representation of a partially ordered set, in this case based on the positive divisors of 24 and 54. After finding the divisors of both numbers, we create a Hasse diagram by placing numbers as points in such a way that each number is connected to its direct divisors with upward lines, representing the divisibility relation.
Step-by-step explanation:
The task is to draw a Hasse diagram for the set of positive divisors of 24 and 54 combined. First, let's find the sets of divisors. For 24, these are A = {1, 2, 3, 4, 6, 8, 12, 24}. For 54, these are B = {1, 2, 3, 6, 9, 18, 27, 54}. Now, we take the union of both sets to get AÙB = {1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 27, 54}.
A Hasse diagram is a graphical representation of the partially ordered set. In this diagram, elements are represented by points, and we draw lines (or edges) upward from smaller to larger elements with respect to the divisibility relation without including transitive edges. Start by placing the smallest element, which is 1, at the bottom. As we move up, numbers that are divisible by other numbers are placed above them.
Here's one possible configuration for the diagram:
- 1 at the bottom since it divides everything,
- 2 just above 1, and similarly 3,
- 4 above 2, 6 above both 2 and 3,
- 8 above 4, 9 above 3,
- 12 above both 4 and 6,
- 18 above both 6 and 9,
- 24 above both 8 and 12,
- 27 above 9,
- 54 at the top since it's divisible by 2, 3, 6, 9, 18, and 27.
Each number is connected by a line to its divisors without skipping any intermediate divisors to respect the partial order defined by divisibility.