Determine whether the following posets are lattices:
i) {1, 2, 3, 4, 5}. ii) {1, 2, 4, 8, 16}

To determine whether the given posets (partially ordered sets) are lattices, we need to check if, for any two elements \(a\) and \(b\) in the poset, both the **least upper bound (supremum)** and the **greatest lower bound (infimum)** exist.

### i) Poset: \(\{1, 2, 3, 4, 5\}\)
Assume the elements are ordered by divisibility (i.e., \(a \leq b\) if \(a\) divides \(b\)).

- **Comparing pairs of elements**:
  - \(1\) divides all other elements, so it is the least element.
  - \(5\) is prime and doesn't divide any other element except itself.
  - For any two elements, check their greatest common divisor (gcd) as the greatest lower bound (infimum) and least common multiple (lcm) as the least upper bound (supremum).

For pairs like \((2, 3)\), \((2, 5)\), \((3, 5)\), etc., there is no common divisor greater than 1 (their gcd is 1), but the lcm is not an element of the set.

- **Conclusion**:
  - This poset is **not a lattice** because not all pairs have both a supremum and infimum within the set.

### ii) Poset: \(\{1, 2, 4, 8, 16\}\)
Assume the elements are ordered by divisibility.

- **Comparing pairs of elements**:
  - Each pair of elements has a greatest lower bound (gcd) within the set and a least upper bound (lcm) also within the set.
  - For example, for \(a = 4\) and \(b = 8\):
    - \(\text{gcd}(4, 8) = 4\), which is in the set.
    - \(\text{lcm}(4, 8) = 8\), which is also in the set.

- **Conclusion**:
  - This poset **is a lattice** because for every pair of elements, both the supremum and infimum exist within the set.

### Summary:
- i) \(\{1, 2, 3, 4, 5\}\) **is not a lattice**.
- ii) \(\{1, 2, 4, 8, 16\}\) **is a lattice**.

Would you like to explore the concepts of lattice properties further or need any clarifications?

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1. What are the properties of a lattice?
2. How do gcd and lcm relate to lattices in posets?
3. What is the significance of ordering by divisibility in posets?
4. Can a poset be a lattice if it has elements with no common divisors?
5. How do we find the supremum and infimum in different types of posets?

**Tip**: When dealing with lattices in posets, always verify that both the least upper bound and greatest lower bound exist for all pairs of elements in the set.

Explore examples of posets and their classification as lattices or non-lattices. Learn about gcd, lcm, and the significance of ordering by divisibility in posets. Detailed solutions provided for {1, 2, 3, 4, 5} and {1, 2, 4, 8, 16}.

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