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Properties of generalized union and intersection. Property 1: For every ( A t ) t 2 T : 1) ∀ t ∈ T A t ⊂ S t 2 T A t ∧ T t 2 T A t ⊂ A t ; 2) ∀ t ∈ T A t ⊂ A ⇒ S t 2 T A t ⊂ A ; 3) ∀ t ∈ T A ⊂ A t ⇒ A ⊂ T t 2 T A t ; 4) A ∪ ( S t 2 T A t ) = S t 2 T ( A ∪ A t ); A ∩ ( S t 2 T A t ) = S t 2 T ( A ∩ A t ); 5) A ∩ ( T t 2 T A t ) = T t 2 T ( A ∩ A t ); A ∪ ( T t 2 T A t ) = T t 2 T ( A ∪ A t ) . Sets Property 2: For every ( A t ) t 2 T and ( B t ) t 2 T : S t 2 T A t ∪ S t 2 T B t = S t 2 T ( A t ∪ B t ) ; T t 2 T A t ∩ T t 2 T B t = T t 2 T ( A t ∩ B t ) . But: S t 2 T ( A t ∩ B t ) ⊂ S t 2 T A t ∩ S t 2 T B t ; T t 2 T A t ∪ T t 2 T B t ⊂ T t 2 T ( A t ∪ B t ) . Property 3 (Generalized De Morgan's Laws): For every ( A t ) t 2 T : A \ S t 2 T A t = T t 2 T ( A \ A t ) (so: S t 2 T A t = T t 2 T A t ) ; A \ T t 2 T A t = S t 2 T ( A \ A t ) (so: T t 2 T A t = S t 2 T A t ) .
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