Hi, i should be this proof

 

Let [latex]R[/latex] be an ordering of [latex]A[/latex]. Prove that [latex]R^{-1}[/latex] is also an ordering of [latex]A[/latex], and for [latex]B\subset{A}[/latex],

 

(A) [latex]a[/latex] is the least element of [latex]B[/latex] in [latex]R^{-1}[/latex] if and only if [latex]a[/latex] is the greatest element of [latex]B[/latex] in [latex]R[/latex].

 

(B) Similarly for (minimal and maximal) and (supremum and infimum.

 

Good look!

Hi, i should be this proof

 

Let [latex]R[/latex] be an ordering of [latex]A[/latex]. Prove that [latex]R^{-1}[/latex] is also an ordering of [latex]A[/latex], and for [latex]B\subset{A}[/latex],

 

(A) [latex]a[/latex] is the least element of [latex]B[/latex] in [latex]R^{-1}[/latex] if and only if [latex]a[/latex] is the greatest element of [latex]B[/latex] in [latex]R[/latex].

 

(B) Similarly for (minimal and maximal) and (supremum and infimum.

 

Good look!

 

This is quite clearly a homework problem. You need to show a reasonable attempt before we will help you.

Yo have reason, in the first i think as follow

 

(a) Proof

 

Assume that [latex]a[/latex] is the least element of [latex]B[/latex] in [latex]R^{-1}[/latex], then

 

[latex]\forall x\in{B},a\leq{x}[/latex]

 

Now, for [latex]R[/latex] is

 

[latex]\forall x\in{B},x\leq{a}[/latex]

 

Hence [latex]a[/latex] is the greatest element of [latex]B[/latex] in [latex]R[/latex]. The reciprocal proof is similary...

 

(b) Proof

 

For Supremum...

 

Assume that [latex]a[/latex] is the supremum of [latex]B[/latex] in [latex]R^{-1}[/latex], then

 

[latex]\forall x\in{B},x\leq{a}[/latex]

 

For R is

 

[latex]\forall x\in{B},a\leq{x}[/latex]

 

This prove that [latex]a[/latex] is lower bound of [latex]B[/latex] in [latex]R[/latex]. Now only remains to prove that [latex]a[/latex] is the mininum lower bound...

 

How i prove this?

 

Good day

Now only remains to prove that [latex]a[/latex] is the mininum lower bound...

 

How i prove this?

 

Good day

 

Use the fundamental definition.