Let T be a countable set, A = { A ⊆ T : A is finite or A c is finite } and μ : A → [ 0 , ∞ ) definided for every A ∈ A by μ ( A ) = { 0 , if A is finite 1 , if A c is finite. Then every set A ∈ A , such that A c is finite, is an atom with respect of μ.But I can't see how it is concluded that every set A ∈ A , such that A c is finite, is an atom with respect of μ, can someone help me?

Question

Let   T be a countable set,       A    =  {  A  ⊆  T  :  A   is finite or       A    c     is finite  } and   μ  :      A    →  [  0  ,  ∞  ) definided for every   A  ∈      A   by  μ  (  A  )  =      {                            0          ,                           if           A           is finite                                       1          ,                           if                       A                          c                                 is finite.                         Then every set   A  ∈      A  , such that       A    c   is finite, is an atom with respect of   μ.But I can&#039;t see how it is concluded that every set   A  ∈      A  , such that       A    c   is finite, is an atom with respect of   μ, can someone help me?

Maggie Bowman · Accepted Answer

If       A    c   is finite, then A must be infinite. Otherwise,   T  =  A  ∪      A    c   would be finite. So   μ  (  A  )  =  1. Now, if   E  ⊆  A and   E  ∈      A  , then, by the latter condition, E is finite and satisfies   μ  (  E  )  =  0, or       E    c   is finite, which , again. implies that E is infinite and   μ  (  E  ). So either   μ  (  E  )  =  μ  (  A  ) or   μ  (  E  )  =  0 for all   E  ⊆  A with   E  ∈      A  , which is exactly the definition of A being an atom.

Let T be a countable set, <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="script">A

Answered question

Answer & Explanation

New Questions in Pre-Algebra