Abstract Algebra/Sets

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[edit] Sets

In the so called naive set theory, which is sufficient for our purposes, the notion of a set is not clearly defined.

All we can say is that for an element $x$ and a set $A$ , either $x \in A$ or $x \not\in A$ .

There is also an empty set, written $\emptyset$ , such that $\forall x, x \not \in \emptyset$ . The empty set can be shown to be unique.

[edit] Comprehensive notation

If it is not possible to list the elements of a set, it can be defined by giving a property that its elements are sole to possess.

[edit] Example

$\mathbb{Z}$ is the set of integers. The set of even integers could be written $\{x \in \mathbb{Z} : x \mbox{ is even}\}$ .

The colon (:) here is read as "such that". The vertical bar (|) is synonymous. This notation will come up a lot in the rest of this book, so it is important for the reader to familiarize themselves with this.

[edit] Number Sets

Naturals: $\mathbb{N} := \{0, 1, 2, \ldots\}$
Integers: $\mathbb{Z} := \{0, \pm 1, \pm 2, \ldots\}$
Rationals: $\mathbb{Q} := \left\{\frac{p}{q} : p \in \mathbb{Z} \mbox{ and } q \in \mathbb{N} \setminus \{0\}\right\}$
Real : $\mathbb{R}$
Complex: $\mathbb{C}$

[edit] Relations

For sets $A$ and $B$ , we define set inclusion as follows:
$A \subseteq B :\Leftrightarrow x \in A \rightarrow x \in B$

By the axiom of extensionnality, $A = B \Leftrightarrow A \subseteq B \mbox{ and } B \subseteq A$ .

[edit] Operations

For sets $A$ and $B$ , we define the following operations:
Intersection: $A \cap B := \{ x \mid x \in A \mbox{ and } x \in B\}$
Union: $A \cup B := \{ x \mid x \in A \mbox{ or } x \in B\}$
Cartesian product: $A \times B := \{(x,y) \mid x \in A \mbox{ and } y \in B\}$
Difference: $A \setminus B := \{ x \mid x \in A \mbox{ and } x \not\in B\}$
Symmetric difference: $A \bigtriangleup B := (A \setminus B) \cup (B \setminus A)$ The text in its current form is incomplete.