Prove Let R be a partial order for a set A Let B be a subset

Prove: Let R be a partial order for a set A. Let B be a subset of A. Then if infimum of B exists, it is unique.

Solution

Answer:

Let R be a partial order for a set A. Let B be a subset of A

Suppose that c and d are both infimums for the subset B. c and d are lower bounds of B. Since c is a inf(B) and d is an lower bound, c R d. Since d is a inf(B) and c is an lower bound, d R c.

Since R is partial order on A implies R is antisymmetric, so, c R d and d R c c = d. Thus, inf(B), if it exists, is unique