TLA + "Specifying Systems" draft translation --Section 1.2 Sets (collections)

This section points

• The basic concept of the collection: elements, relationships $\in$ , equal conditions;
• A set of four basic operations: $\cap$ intersection, $\cup$ union, $\subseteq$ subset and $\ setminus$ difference sets;

Set theory is the basis of the general mathematics. Often described as the set of elements of the "aggregation", but said the collection is not particularly exact polymerization, the concept of the collection is so fundamental that we not to define it. We first set and relationships $\in$ as undefined concepts, $x \in S$ represents $x$ is $S$ an element, we often say, "in the collection $S$ in "(is in), rather than" is $S$ a element "(is a element of).

A collection can have a limited number of elements that can have an unlimited number of elements. The set of all natural numbers (0,1,2, etc.) is an infinite set. Set of natural numbers smaller than 3 is composed of a finite set, including 0,1,2 these three elements, we can do it in mind $\{0,1,2\}$.

A collection depends entirely on its elements. Two sets are equal if and only if they have the same elements. therefore, $\{0,1,2\}$, $\{2,1,0\}$, $\{0,0,1,2,2\}$ is the same set, i.e., has a set of elements of the three elements 0,1,2. Empty set, denoted $\{\}$ , It is a unique set has zero elements.

Common set as follows:
$\cap$ intersection $\qquad\cup$ union $\qquad\subseteq$ child Collection $\ Qquad \ setminus$ difference set

Here is the definition and use examples:

• $S\cap T$ $S$ and $T$ set contains elements
  $\{1,-1/2,3\}\cap \{1,2,3,5,7\}=\{1,3\}$
• $S\cup T$ $S$ or $T$ collection contains the elements (which may contain)
  $\{1,-1/2\}\cup \{1,5,7\}=\{1,1/2,5,7\}$
• $S\subseteq T$ is TRUE if and only if $S$ The elements are also $T$ elements in the
  $\{1,3\}\subseteq \{3,2,1\}$
• $S\setminus T$ in $S$ , but not in $T$ set of elements
  $\{1,-1/2,3\}\cup \{1,5,7\}=\{-1/2,3\}$

In that before starting to learn how to define the system, we need to know all the information about the set up. We will continue to discuss the set theory in Section 6.1.