Consider the set Pa(1,2,3,4,5) under the partial ordering R=((1, 1), (1,2), (1,3), (1,4), (1,5),(2,2)(2,3),(2,5),(3,3),(3,5)(4,4),(4,5),(5,5)). The minimum number of ordered pairs that need to be added to R to make (P,R) a lattice is zero.

Let's break down the problem.

We are given a set \( P = \{1, 2, 3, 4, 5\} \) and a partial ordering \( R \) consisting of the following relations:

\[
R = \{(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (2, 2), (2, 3), (2, 5), (3, 3), (3, 5), (4, 4), (4, 5), (5, 5)\}
\]

### Key Definitions:

1. **Partial Ordering**: A partial order is a binary relation that is reflexive, antisymmetric, and transitive.
2. **Lattice**: A lattice is a partially ordered set in which every pair of elements has both a **least upper bound (join)** and a **greatest lower bound (meet)**.

### Step-by-step analysis:

1. **Check the partial order properties:**
   - Reflexivity: Each element is related to itself, as we see \( (1, 1), (2, 2), (3, 3), (4, 4), (5, 5) \).
   - Antisymmetry: If \( a \leq b \) and \( b \leq a \), then \( a = b \). The given pairs comply with this.
   - Transitivity: If \( a \leq b \) and \( b \leq c \), then \( a \leq c \). Based on the given pairs, this holds (e.g., \( 1 \leq 2 \) and \( 2 \leq 3 \) implies \( 1 \leq 3 \)).

2. **Check if the relation defines a lattice**:
   - **Greatest Lower Bound (GLB)** or **meet**: For every pair of elements \( a \) and \( b \), there should be a greatest element \( c \) such that \( c \leq a \) and \( c \leq b \).
   - **Least Upper Bound (LUB)** or **join**: For every pair of elements \( a \) and \( b \), there should be a least element \( d \) such that \( a \leq d \) and \( b \leq d \).

To form a lattice, we need to ensure that for every pair of elements in the set \( P \), both the **meet** and **join** exist.

3. **Examine the structure**:
   The given ordering of elements seems to already define a structure where:
   - 1 is the least element (since \( 1 \leq 2 \), \( 1 \leq 3 \), \( 1 \leq 4 \), \( 1 \leq 5 \)).
   - 5 is the greatest element (since \( 2 \leq 5 \), \( 3 \leq 5 \), \( 4 \leq 5 \)).
   - Intermediate relationships such as \( 2 \leq 3 \leq 5 \) and \( 4 \leq 5 \) further support this.

Thus, this structure already allows for the meet and join for any pair of elements. Therefore, **no additional pairs need to be added**.

### Conclusion:
The minimum number of ordered pairs that need to be added to \( R \) to make \( (P, R) \) a lattice is indeed zero. This means that \( (P, R) \) is already a lattice.

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Would you like more details or have any questions?  
Here are 5 related questions to explore further:

1. What is the difference between a partial order and a total order?
2. How can we construct the Hasse diagram of the given relation \( R \)?
3. What conditions must a partially ordered set satisfy to be considered a lattice?
4. Can a lattice be finite or infinite, and how does that affect its structure?
5. How do meet and join operations behave in lattices with more complex sets?

**Tip**: Always verify reflexivity, antisymmetry, and transitivity when checking if a relation is a partial order.

Explore the problem of determining if a partial ordering R on set P = {1, 2, 3, 4, 5} forms a lattice. Analyze the structure of the relations and verify the lattice conditions using the meet and join operations. Suitable for college-level discrete mathematics students.

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