I am reading one book and the author presents what he calls a distributivity formula: ⋃ j ∈ J ⋂ i ∈ I j F i j = ⋂ { i j } ∈ K ⋃ j ∈ J F i j j where K = Π j ∈ J I j (the set of all sequences { i j , j ∈ J}).My question is regarding the notation. The LHS is quite clear to me. However, I'm now sure what are actually the members of K. Say J = { 1 , 2 , … } and I j = { 1 , 2 , … }. What is actually i j here?

Question

I am reading one book and the author presents what he calls a distributivity formula:      ⋃          j      ∈      J            ⋂          i      ∈              I        j                  F    i    j    =      ⋂          {              i        j            }      ∈      K            ⋃          j      ∈      J            F                  i        j              j  where   K  =      Π          j      ∈      J            I    j   (the set of all sequences {      i    j    ,  j  ∈  J}).My question is regarding the notation. The LHS is quite clear to me. However, I&#039;m now sure what are actually the members of K. Say   J  =  {  1  ,  2  ,  …  } and       I    j    =  {  1  ,  2  ,  …  }. What is actually       i    j   here?

Marisol Morton · Accepted Answer

On the RHS, we are finding the intersection of       ⋃          j      ∈      J            F                  i        j              j  . Here, the       i    j   is the values for which we are intersecting; it says   {      i    j    }  ∈  K. If   J  =  {  1  ,  2  ,  3  ,  ⋯  }, then   K  =      I    1    ×      I    2    ×      I    3    ×  ⋯

I am reading one book and the author presents what he calls a distributivity formula: <munder>

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