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Math Help - set theory-partition

  1. #1
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    set theory-partition

    the family set of{Ai,iEI}is a partition of A,
    the family set of{Bj,jEJ}is a partition of B,
    than prove {AixBj,iEI and jEJ}is a parititon of AxB.
    thanks for your helps.
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  2. #2
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    First we show that each cell is not empty.
    Given A_c \times B_d because of partitions \left( {\exists x \in A_{\,c} } \right)\left( {\exists y \in B_d } \right) \Rightarrow \quad \left( {x,y} \right) \in A_c \times B_d .
    Does that prove the first requirement?

    Now disjoint cells.
    If \left( {x,y} \right) \in \left( {A_c \times B_d } \right) \cap \left( {A_j \times B_k } \right) then x \in A_c \cap A_j \,\& \,y \in \left( {B_d \times B_k } \right)<br />
WHY?
    WHY does mean that \left( {A_c = A_j } \right)\,\& \,\left( {B_d = B_k } \right)?
    Now can you explain how this shows that different cells are disjoint?

    Now the covering property.
    If \left( {a,b} \right) \in \left( {A \times B} \right) \Rightarrow \quad \left( {\exists j} \right)\left[ {a \in A_j } \right]\,\& \,\left( {\exists k} \right)\left[ {b \in B_k } \right]\,. How are we sure of this?
    This means that \left( {a,b} \right) \in \left( {A_j \times B_k } \right).
    How does that show that \left( {A \times B} \right) = \bigcup\limits_{j,k} {\left( {A_j \times B_k } \right)}?
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  3. #3
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    thanks ı call you as saint plato thanks for your helps
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