Transitivity and Ordered Fields. How do you prove this? Please help.
In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
prove that only one of the following can be true:
x<y,x=y,y<x
Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
prove that only one of the following can be true:
x<y,x=y,y<x
Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
there are a couple of ways to define an ordered field. which one are you using?
In an ordered field, S, with positive elements, P, for all x,y elements of S define x<y if y-x is an element of P,
prove that only one of the following can be true:
x<y,x=y,y<x
Also prove that if x,y,z are arbitrary and x<y and y<z then x<z.
The set satisfied that given we exactly one of following: . Also given we have . Given, consider . If then we are done. Otherwise xor . Thus, xor i.e. xor .
To prove the other statement, notice that if and then so .