This exercise provides the steps necessary to prove that every partial ordering is i...

This exercise provides the steps necessary to prove that every partial ordering is in a sense the same as the set inclusion relation on a collection of subsets of a set. Let A be a set with a partial order R. For each a ∈ A, let and thus may be partially ordered by ⊆.

(a) Show that if a R b, then S_a ⊆ S_b

(b) Show that if Sa ⊆ Sb, then a R b

(c) Show that for every b ∈ A, an immediate predecessor of b in A corresponds to an immediate predecessor of

(d) Show that if B ⊆ A and x is the least upper bound for B, then S_x is the least upper bound for{S_b: b ∈ B}.