Union and intersection presented in the previous sections were two-argument operations. They can be generalized in a very simple way for arbitrary finite or infinite number of arguments.

Obviously, if the set of indexes contains two elements only, for example I = {1, 2}, then the generalized union is exactly the union of sets A₁ and A₂,

When all sets of the family A are taken from the same universe U, then the generalized De Morgan's laws hold as it is presented in the following theorem 1.6.1.

Proof.

Ad. (a) Suppose, x belongs to the left-hand side of the equality (a). By Definition 1.6.1, x does not belong to any set A_i for i ∈ I. Thus x is in complement -A_i for every i ∈ I and, therefore, it is in the generalized intersection ⎧⎫ _{i ∈ I} (- A_i ).

Conversely, if x is in ⎧⎫ _{i ∈ I} (- A_i ) then it is in all sets (- A_i ) for i ∈ I. By definition of complement, x does not belong to any set of the family A, and, therefore, it does not belong to the generalized union of the family A. Hence x is in - ⎩⎭_{i ∈ I} A_i.

Ad. (b) If x is in - ⎧⎫ _{i ∈ I} A_i , then it is not in the set ⎧⎫ _{i ∈I} A_i. By Definition 1.6.1 for some particular k, x ∉ A_k. Hence x ∈ -A_k, and therefore x ∈ ⎩⎭ _{i ∈ I} (- A_i ).

Conversely, if x is in the generalized union ⎩⎭ _{i ∈ I} (- A_i ) then x must be in at least one set of the family {- A_i : i ∈ I} and therefore is not in at least one of the set A_i . As a consequence, x does not belong to the generalized intersection of the family A and hence x is in its complement.

Question 1.6.1 Let {A_i : i ∈I} be a family of subsets of a given universe, indexed by a set I .

Which of the following properties are valid?

(1) For every k ∈ I, A_k ⊆ ⎩⎭ _{i ∈ I} A_i.

(2) For every k ∈ I, A_k ⊆ ⎧⎫ _{i ∈ I} A_i.

(3) For every k ∈ I, ⎩⎭ _{i ∈ I} A_i ⊆ A_k.

(4) For every k ∈ I, ⎧⎫ _{i ∈ I} A_i ⊆ A_k.

(5) ⎧⎫ _{i ∈ I} A_i ⊆ ⎩⎭ _{i ∈ I} A_i.

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