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Реални броеви

Module by: Liljana Stefanovska. E-mail the author

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Summary: Се воведува множеството од реални броеви и неговиот кардинален број.

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МНОЖЕСТВО РЕАЛНИ БРОЕВИ

Унијата на рационалните и ирационалните броеви го дава множеството реални броеви R, т.е.

R = Q ∪ I . R = Q ∪ I . size 12{R=Q union I "." } {}

Реални броеви има “повеќе” од природни. Кантор (Cantor 1845-1918) докажал дека реалните броеви не може да се подредат во низа, бидејќи секогаш ќе има реални броеви кои не се опфатени во низата. Затоа за реалните броеви се вели дека имаат моќ на континуум. Таква моќ има и множеството на точки од правата (или било која отсечка) како и множеството на точки од рамнината. Не е познато множество со моќ меѓу преброиво и континуум, но постои множество со моќ поголема од континуум. Такво е на пр. множеството од сите реални функции дефинирани на отсечката [0, 1].

Забелешка

Во проширеното множество реални броеви (R∪{−∞,+∞}),(R∪{−∞,+∞}), size 12{ \( R union lbrace - infinity ,`+ infinity rbrace \) ,} {}∀a∈R∀a∈R size 12{ forall a in R} {} важи:

  • a+(+∞)=+∞a+(+∞)=+∞ size 12{a+ \( + infinity \) "=+" infinity } {}, a+(−∞)=−∞a+(−∞)=−∞ size 12{a+ \( - infinity \) = - infinity } {}
  • a⋅(+∞)=+∞a⋅(+∞)=+∞ size 12{a cdot \( + infinity \) "=+" infinity } {}, a⋅(−∞)=−∞a⋅(−∞)=−∞ size 12{a cdot \( - infinity \) = - infinity } {}, за (a>0)(a>0) size 12{ \( a>0 \) } {}
  • a+∞=a−∞=0a+∞=a−∞=0 size 12{ { {a} over {+ infinity } } = { {a} over { - infinity } } =0} {}, за (a≠0)(a≠0) size 12{ \( a <> 0 \) } {}
  • 0a=00a=0 size 12{ { {0} over {a} } =0} {}, за (a≠0)(a≠0) size 12{ \( a <> 0 \) } {}
  • a0=+∞a0=+∞ size 12{ { {a} over {0} } "=+" infinity } {}, за (a>0)(a>0) size 12{ \( a>0 \) } {}
  • ( + ∞ ) + ( + ∞ ) =+ ∞ ( + ∞ ) + ( + ∞ ) =+ ∞ size 12{ \( + infinity \) + \( + infinity \) "=+" infinity } {}
  • ( − ∞ ) + ( − ∞ ) = − ∞ ( − ∞ ) + ( − ∞ ) = − ∞ size 12{ \( - infinity \) + \( - infinity \) = - infinity } {}
  • ( + ∞ ) ⋅ ( + ∞ ) =+ ∞ ( + ∞ ) ⋅ ( + ∞ ) =+ ∞ size 12{ \( + infinity \) cdot \( + infinity \) "=+" infinity } {}
  • ( − ∞ ) ⋅ ( − ∞ ) =+ ∞ ( − ∞ ) ⋅ ( − ∞ ) =+ ∞ size 12{ \( - infinity \) cdot \( - infinity \) "=+" infinity } {}

Изразите 00,±∞±∞,0⋅(±∞),(+∞)+(−∞)00,±∞±∞,0⋅(±∞),(+∞)+(−∞) size 12{ { {0} over {0} } ,~ { { +- infinity } over { +- infinity } } ,~0 cdot \( +- infinity \) ,~ \( + infinity \) + \( - infinity \) } {} не се определени.

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