(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Suppose X and Y are sets. Let P be all pairs (A,f) where A is a subset of X and f is a function from X to Y. Then P is a poset with the relation (A,f)=<(B,g) iff A is a subset of B and f is the restriction of g to A.

Show that if C={(Ai,fi)|i in I} is a chain in P, there is a unique function f:U Ai -> Y such that for each i, fi is the restriction of f that Ai.

2. Relevant equations

A chain is subset of P such that for all elements p,q in the chain p=<q or q=<p.

3. The attempt at a solution

lets start out with (A1,f1)

f1:A1->Y = (f2:A2->Y restricted to A1)

and so

f1: A1->Y = (fn:An->Y restricted to the intersection of all Ai's)

but "restricted to the intersection of all Ai's" is the same as "restricted to A1", or more generally to Aj, the jth member of the chain. This is true because there are all inside eachother by definition of the poset.

so would fn:An->Y fit the role of the fuction we were looking for?

would therefore (An,fn) be the upper bound asked for?

It feels too simple to be the right answer.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Exercise on Posets, Chains and upper bounds

**Physics Forums | Science Articles, Homework Help, Discussion**