To determine whether the partial ordered set is({1,2,4,8,16},\)
kirala gayathri
Answered question
2022-09-02
To determine whether the partial ordered set is({1,2,4,8,16},\) lattice or not
Answer & Explanation
xleb123
Skilled2023-06-03Added 181 answers
To determine whether the partially ordered set ({1, 2, 4, 8, 16}) forms a lattice or not, we need to check if it satisfies the properties of a lattice.
A lattice is a partially ordered set in which every pair of elements has both a supremum (least upper bound) and an infimum (greatest lower bound).
Let's examine the given set and determine if it meets these conditions.
**Step 1: Check for the existence of a supremum for every pair of elements.**
We need to find the least upper bound (supremum) for each pair of elements in the set.
For example, let's consider the pair (2, 4). The supremum of this pair would be the smallest element that is greater than or equal to both 2 and 4. In this case, we can see that 4 is the smallest element that satisfies this condition.
Similarly, for the pairs (2, 8), (2, 16), (4, 8), (4, 16), and (8, 16), we can observe that the supremums are 8, 16, 16, 8, and 16, respectively.
**Step 2: Check for the existence of an infimum for every pair of elements.**
Next, we need to find the greatest lower bound (infimum) for each pair of elements in the set.
Taking the same pairs we considered before, we can find the infimums:
- For (2, 4), the infimum is 2.
- For (2, 8), the infimum is 2.
- For (2, 16), the infimum is 2.
- For (4, 8), the infimum is 4.
- For (4, 16), the infimum is 4.
- For (8, 16), the infimum is 8.
**Step 3: Verify if all supremums and infimums exist in the set.**
To determine if the set forms a lattice, we need to check if all the supremums and infimums we found in Steps 1 and 2 actually exist in the given set.
In this case, all the supremums (4, 8, 16) and infimums (2, 4, 8) we identified are present in the set ({1, 2, 4, 8, 16}).
Therefore, we can conclude that the partially ordered set ({1, 2, 4, 8, 16}) forms a lattice.
I hope this explanation clarifies the concept of a lattice and how to determine if a given partially ordered set is a lattice.
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