Darío Posted February 29, 2012 Posted February 29, 2012 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!
DrRocket Posted February 29, 2012 Posted February 29, 2012 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.
Darío Posted March 2, 2012 Author Posted March 2, 2012 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
DrRocket Posted March 2, 2012 Posted March 2, 2012 Now only remains to prove that [latex]a[/latex] is the mininum lower bound... How i prove this? Good day Use the fundamental definition.
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