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 ...........\_\_\_.

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.

Answer 1

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: \[ \boxed{5} \]