# Set System

A Set System is a Formal System for sets and Set Operations.

**AKA:**Set Theory, Set Formal System, Formal Set System, Set Algebra, Algebra of Sets, [math](V,\mathcal{I})[/math].**Context:**- It must include Set Operations (that can be constrained by Set Axioms).
- It must include a Set Formal Language.
- It can range from being an Independent Set System to being a Dependent System System.

**Counter-Example(s):****See:**Multiset, Rough Set Theory, Infinite Set, Empty Set, Countable Set, Matroid, Discrete Mathematics.

## References

### 2015

- (Wikipedia, 2015) ⇒ http://en.wikipedia.org/wiki/Naive_set_theory
- Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. [1] The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of set theory concepts in most contemporary mathematics.
Sets are of great importance in mathematics; in fact, in modern formal treatments, most mathematical objects (numbers, relations, functions, etc.) are defined in terms of sets. Naive set theory can be seen as a stepping-stone to more formal treatments, and suffices for many purposes.

- Naive set theory is one of several theories of sets used in the discussion of the foundations of mathematics. [1] The informal content of this naive set theory supports both the aspects of mathematical sets familiar in discrete mathematics (for example Venn diagrams and symbolic reasoning about their Boolean algebra), and the everyday usage of set theory concepts in most contemporary mathematics.

### 2009

- (Wikipedia, 2009) ⇒ http://en.wikipedia.org/wiki/Algebra_of_sets
- The algebra of sets develops and describes the basic properties and laws of sets, the set-theoretic operations of union, intersection, and complementation and the relations of set equality (mathematics) and set inclusion. It also provides systematic procedures for evaluating expressions, and performing calculations, involving these operations and relations.
**The fundamental laws of set algebra:**The binary operations of set union and intersection satisfy many identities. Several of these identities or "laws" have well established names. Three pairs of laws, are stated, without proof, in the following proposition.

```
PROPOSITION 1: For any sets A, B, and C, the following identities hold:
```

commutative laws:

* A \cup B = B \cup A\,\!
* A \cap B = B \cap A\,\!

associative laws:

* (A \cup B) \cup C = A \cup (B \cup C)\,\!
* (A \cap B) \cap C = A \cap (B \cap C)\,\!

distributive laws:

* A \cup (B \cap C) = (A \cup B) \cap (A \cup C)\,\!
* A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\,\!

Notice that the analogy between unions and intersections of sets, and addition and multiplication of numbers, is quite striking. Like addition and multiplication, the operations of union and intersection are commutative and associative, and intersection distributes over unions. However, unlike addition and multiplication, union also distributes over intersection.

The next proposition, states two additional pairs of laws involving three specials sets: the empty set, the universal set and the complement of a set.

PROPOSITION 2: For any subset A of universal set U, where Ø is the empty set, the following identities hold:

identity laws:

* A \cup \varnothing = A\,\!
* A \cap U = A\,\!

complement laws:

* A \cup A^C = U\,\!
* A \cap A^C = \varnothing\,\!

```
```

### 1966

- Paul J Cohen. (1966). “Set Theory and the Continuum Hypothesis." W. A. Benjamin, Inc.