Question

Consider the set T whose elements are all the factors of 1000 and denote a relation τ ⊂ T × T such that aτ b if a is a factor of b. i. Verify that (T, τ ) is a poset. ii. Give the least upper bound for 8 and 25, as well as one other upper bound (< 1000). iii. Give the greatest lower bound for 20 and 50, as well as any other lower bound (> 1). iv. Is (T, τ ) a lattice? Explain why/why not

Answer 1

Here are the findings. (i) Yes, (T, τ) is a poset. (ii)The least upper bound for 8 and 25 is 200, and another upper bound (< 1000) is 500. (iii)The greatest lower bound for 20 and 50 is 10, and another lower bound (> 1) is 1. (iv)Yes, (T, τ) is a lattice.

Step-by-step explanation:

i. To verify that (T, τ) is a poset, we need to check reflexivity, antisymmetry, and transitivity. Reflexivity holds since every element is related to itself through τ. Antisymmetry is satisfied as if aτb and bτa, then a must equal b. Transitivity holds, as if aτb and bτc, then aτc. Thus, (T, τ) is a poset.

ii. To find the least upper bound for 8 and 25, we need to identify their common factors. The factors of 8 are 1, 2, and 4, and the factors of 25 are 1 and 5. The least common factor is 1, and the least upper bound is found by multiplying this common factor with the larger number, i.e., 1 * 25 = 25. Another upper bound (< 1000) can be found by selecting a common factor like 2, resulting in 2 * 25 = 50.

iii. For the greatest lower bound of 20 and 50, we identify their common factors. The factors of 20 are 1, 2, 4, 5, 10, and 20, and the factors of 50 are 1, 2, 5, 10, 25, and 50. The greatest common factor is 10, which serves as the greatest lower bound. Another lower bound (> 1) can be any common factor, e.g., 2, resulting in a lower bound of 2.

iv. A lattice requires a poset where every pair of elements has both a least upper bound (LUB) and a greatest lower bound (GLB). In this case, we've identified LUBs and GLBs for various pairs, satisfying the lattice properties. Thus, (T, τ) is a lattice.