[¤j¾Ç]2ÃDÃÒ©ú

[¤j¾Ç]2ÃDÃÒ©ú

¥Ñ ³X«È ©ó ¬P´Á¤é ¤Q¤ë 02, 2011 4:59 pm


1.
Let A be a non-empty subset of R. Define -A := {-X I X‘kA}
Show the following statements.
(a) A has a supremum if and only if -A has an ifimum, in
which case we have inf(-A) = -supA.
(b) A has an in mum if and only if -A has a supremum, in
which case we have sup(-A) = -inf A.

2.
Show that the completeness axiom of real number system (i.e.
the Least Upper Bound Property) is equivalent to the Greatest
Lower Bound Property: Every non-empty set A of real numbers
that has a lower bound has a greatest lower bound.
»P¤W¤@ÃD¦³Ãö

³X«È

 

¥Ñ Tzwan ©ó ¬P´Á¤» ¤»¤ë 08, 2013 4:06 pm


1.
  (a)
  ¥ªÁä: ÂIÀ»ÁY©ñ; ¥kÁä: Æ[¬Ý­ì¹Ï
  (b)
  ¥ªÁä: ÂIÀ»ÁY©ñ; ¥kÁä: Æ[¬Ý­ì¹Ï

2.
  ¥ªÁä: ÂIÀ»ÁY©ñ; ¥kÁä: Æ[¬Ý­ì¹Ï

  So the completeness axiom of real number system is equivalent to the Greatest Lower Bound Property.

Tzwan
ªì¾ÇªÌ
ªì¾ÇªÌ
 
¤å³¹: 33
µù¥U®É¶¡: 2013-04-01




¤j¾Ç¥H¤W¼Æ¾Ç°ÝÃD