Let Q denote all the finite subsets of N that have an even n

Let Q denote all the finite subsets of N that have an even number of elements. Consider the partial order on Q given by the relation S T. Let A = {1, 2} and B = {1, 3}. What are all the common upper bounds of A and B in Q? Is there a least upper bound of A and B?

Solution

Definition 5.1.2 Let \'X, ( be a poset and let A X be any subset of X. An element, b X, is a lower bound of A iff b a for all a A. An element, m X, is an upper bound of A iff a m for all a A. An element, b X, is the least element of A iff b A and b a for all a A. An element, m X, is the greatest element of A iff m A and a m for all a A

common upper bound will be 3

lest upper bound will be 1