Question:
Let \(P\) be the partial order defined on the set \(\{1, 2, 3, 4\}\) as follows:
\[
P = \{(x, x) \mid x \in \{1, 2, 3, 4\}\} \cup \{(1, 2), (3, 2), (3, 4)\}
\]
The number of total orders on \(\{1, 2, 3, 4\}\) that contain \(P\) is ...........\_\_\_.
Let \(P\) be the partial order defined on the set \(\{1, 2, 3, 4\}\) as follows:
\[
P = \{(x, x) \mid x \in \{1, 2, 3, 4\}\} \cup \{(1, 2), (3, 2), (3, 4)\}
\]
The number of total orders on \(\{1, 2, 3, 4\}\) that contain \(P\) is ...........\_\_\_.
Show Hint
To determine the number of total orders from a partial order, analyze the transitive closure and enumerate all valid permutations that satisfy the given relations.
Updated On: Jan 23, 2025
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Solution and Explanation
A total order is an extension of a partial order in which every pair of elements is comparable. Based on the given partial order \(P\), we know the following relations:
\[
1 \leq 2, \quad 3 \leq 2, \quad 3 \leq 4.
\]
Using these relations, the valid total orders containing \(P\) are:
\(1 \leq 3 \leq 4 \leq 2\)
\(1 \leq 3 \leq 2 \leq 4\)
\(1 \leq 4 \leq 3 \leq 2\)
\(3 \leq 1 \leq 4 \leq 2\)
\(3 \leq 1 \leq 2 \leq 4\)
Thus, there are 5 total orders that extend \(P\).
Final Answer:
\[
\boxed{5}
\]
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