Relations: prove exists at most 1 least elt in part. ordered

[emlevy]

For a set X with a relation of partial order R the element x in X is called the least element of X if, for all y in X, x relates to y (xRy). Prove that in a partially ordered set there exists at most one least element.

 

[pka]

Re: Relation help - PLEASE HELP ASAP

If each of $\displaystyle a~\&~b$ is a least element in $\displaystyle X$ then $\displaystyle a \prec b~\&~b \prec a$.
Why is that true? What does that a about $\displaystyle a~\&~b$?

 

[emlevy]

Re: Relation help - PLEASE HELP ASAP

I'm not sure...

 

[pka]

Re: Relation help - PLEASE HELP ASAP

Well do you know the definitions?
Is a partial ordering antisymmetric?

 

[emlevy]

yes it is antisymmetric

 

[pka]

yes it is antisymmetric

What does that mean if $\displaystyle a\prec b~\&~b\prec a$?

$\displaystyle \prec$ means preceeds.