I know induction from my lower division classes, but have no idea what to do when it comes to sets. For example a directed set is such that every two elements has an upperbound. We are looking at power sets specifically with the inherited binary operation as inclusion. Where do I start from here? I could say, well we can't have the empty set since it doesn't qualify(not sure to to reason this well). Does the definition imply I need at least 3 distinct elements? I believe we can assumed inclusion as in a, b are both contained in a bigger subspace c. Do I start off by saying, we have 3 elements such that a,b contained in c. Then c is maximal element. Next, we have 4 elements? Any help? I believe since I'm asked to use induction it implies the set is infinite.
Actually, I also would like clarification on what a maximal element truly means? I always thought it contains the lower sets within it.(My class is using inclusion for the set binary operation)