Question

Problem 4. (Due September 4.) Prove that Z3Z3 is not an ordered field.

Here is a solution for Problem 1.

The reverse (⇐⇐) implication is false. Let X={♠️,♣️,♦️️}X={♠️,♣️,♦️️} and Y={?,?}Y={?,?}. Define ff by f(♠️)=f(♣️)=?andf(♦️)=?..f(♠️)=f(♣️)=?andf(♦️)=?..By checking all four possible pairs of disjoint sets from YY, it is easy to verify the inverse images of disjoint subsets of YY are disjoint in XX, but ff is clearly not injective

Answer 1