MathDB

Problem 6

Part of 2011 VTRMC

Problems(1)

set with asymmetric relation, subset has property

Source: VTRMC 2011 P6

5/14/2021
Let SS be a set with an asymmetric relation <<; this means that if a,b∈Sa,b\in S and a<ba<b, then we do not have b<ab<a. Prove that there exists a set TT containing SS with an asymmetric relation ≺\prec with the property that if a,b∈Sa,b\in S, then a<ba<b if and only if a≺ba\prec b, and if x,y∈Tx,y\in T with x≺yx\prec y, then there exists t∈Tt\in T such that x≺t≺yx\prec t\prec y.
Sets