In book i read now, the author defines a field just like this:A field is a non-empty class of subsets of Ω closed under finite union,finite intersection and complements.In the following, he says like this:A minimal set of postulates for A to be a field is (i) Ω ∈ A .(ii) T ∈ A implies T ¯ ∈ A .(iii) S , T ∈ A implies S ∪ T ∈ AThough I know why it exactly defines a field, the author says nothing about why they are minimal. He even uses the word minimal without an exact definition. Can anyone help me with this problem?

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In book i read now, the author defines a field just like this:A field is a non-empty class of subsets of   Ω closed under finite union,finite intersection and complements.In the following, he says like this:A minimal set of postulates for       A to be a field is (i)  Ω  ∈  A  .(ii)  T  ∈  A implies       T    ¯    ∈  A  .(iii)  S  ,  T  ∈  A implies   S  ∪  T  ∈  AThough I know why it exactly defines a field, the author says nothing about why they are minimal. He even uses the word minimal without an exact definition. Can anyone help me with this problem?

Jameson Freeman · Accepted Answer

The postulates are minimal becasue no two of them implies the third one. To see this we have to produce counter-examples.1. The empty collection satisfies (ii) and (iii) (vacuously) but does not satisfy (i).2.   {      R    ,  [  0  ,  1  ]  ,  [  0  ,  2  ] $ satisfies (i) and (iii) but not (ii)3.   {      R    ,  [  0  ,  1  ]  ,  [  0  ,  2  ]  ,      R    ∖  [  0  ,  1  ]  ,      R    ∖  [  0  ,  2  ]  ,  ∅  } satisfies (i) and (ii) but not (iii)

In book i read now, the author defines a field just like this: A field is a non-empty class of sub

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