Set theory homework Please be descriptive thanks Let x and y

Set theory homework. Please be descriptive. thanks!

Let x and y be linearly ordered sets. Prove that if x and y are isomorphic and x is linearly ordered, then y is linearly ordered.

Solution

In the question only X is required to be linearly ordered. . (If Y is also linearly ordered, then the question is redundant).

Modified question: X and Y are isomorphic sets. If X is linearly ordered, prove that Y is also linearly ordered.

Proof: X and Y isomorphic sets means there exists a bijection f: X to Y (one one onto map).

Given a and b , a not equal to b, in Y, there exists unique c and d in X such that f(c)=a and f(d)=b.

As X is linearly ordered, either c is less than d or d is less than c.

Assume c <d. Declare a <b in Y.

This defines a linear order in Y (Basically we are transferring the linear order on X , by using the bijective map f from X to Y--it is just relabelling elements).

This proves the claim