Question

Consider the ordering relation a | b ⊆ N x N over natural numbers N such that a | b if there exists c belong to N such that a*c=b. Then ___________

(a) | is an equivalence relation

(b) It is a total order

(c) Every subset of N has an upper bound under |

(d) (N,|) is a lattice but not a complete lattice

I have been asked this question by my college professor while I was bunking the class.

My query is from Relations topic in portion Relations of Discrete Mathematics

Answer 1

The correct choice is (d) (N,|) is a lattice but not a complete lattice

To explain: A set is called lattice if every finite subset has a least upper bound and greatest lower bound. It is termed as a complete lattice if every subset has a least upper bound and greatest lower bound. As every subset of this will not have LUB and GLB so (N,|) is a lattice but not a complete lattice.