## Generalized Set Operations

### Subjects to be Learned

• generalized set union
• generalized set intersection
• generalized Cartesian product
• generalized De Morgan's Rules

### Contents

As we saw earlier, union, intersection and Cartesian product of sets are associative. For example
To denote either of these we often use A B C .

This can be generalized for the union of any finite number of sets as A1 A2 .... An ,
which we write as

Ai

This generalized union of sets can be rigorously defined as follows:

Definition ( Ai) :
Basis Clause: For n = 1 ,   Ai = A1.
Inductive Clause:   Ai = ( Ai) An+1

Similarly the generalized intersection Ai and generalized Cartesian product Ai can be defined.

Based on these definitions, De Morgan's law on set union and intersection can also be generalized as follows:

Theorem (Generalized De Morgan)

= ,     and
=

Proof: These can be proven by induction on n and are left as an exercise.

Next -- Recursive Definition of Functions

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