Find the least upper bound and greatest lower bound of \( S = \{X, Y, Z\} \) if they exist, of the poset whose Hasse diagram is shown below:
Find the least upper bound and greatest lower bound of \( S = \{X, Y, Z\} \) if they exist, of the poset whose Hasse diagram is shown below:
Show Hint
- The least upper bound is T and the greatest lower bound is X.
- The least upper bound is Z and the greatest lower bound is E.
- The least upper bound is I and the greatest lower bound is Y.
- The least upper bound is T and the greatest lower bound is Y.
The Correct Option is D
Solution and Explanation
Step 1: Understanding the Hasse diagram.
In a Hasse diagram, the least upper bound (LUB) is the smallest element that is greater than or equal to all elements in the set, and the greatest lower bound (GLB) is the largest element that is less than or equal to all elements in the set.
Step 2: Identify the least upper bound.
The least upper bound of the set \( \{X, Y, Z\} \) is the smallest element that is greater than or equal to both \( X \), \( Y \), and \( Z \). Based on the diagram, the least upper bound is \( T \).
Step 3: Identify the greatest lower bound.
The greatest lower bound of the set \( \{X, Y, Z\} \) is the largest element that is less than or equal to both \( X \), \( Y \), and \( Z \). From the diagram, the greatest lower bound is \( Y \).
Step 4: Conclusion.
Thus, the correct answer is (4) The least upper bound is \( T \) and the greatest lower bound is \( Y \).
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