Set

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
*A*_{1}
*A*_{2}
....
*A*_{n} ,

which we write as

*A*_{i}

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

**Definition (
***A*_{i}) :

**Basis Clause:** For *n = 1* ,
*A*_{i}
** = ***A*_{1}.

**Inductive Clause:**
*A*_{i} = (
*A*_{i})
*A*_{n+1}

Similarly the **generalized intersection**
*A*_{i} and **generalized Cartesian product**
*A*_{i} 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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