Set theory

Basic Definition
The term "set" can be thought as a well-defined collection of objects. In set theory, These objects are often called "elements". (Example)    $$x \in \mathbb{R}$$. In this case, $$x$$ is the element and $$\mathbb{R}$$ is the set of all real numbers.
 * We usually use capital letters for the sets, and lowercase letters for the elements.
 * If an element $$a$$ belongs to a set $$A$$, we can say that "$$a$$ is a member of the set $$A$$", or that "$$a$$ is in $$A$$", or simply write $$ a \in A $$.
 * Similarly, if $$a$$ is not in $$A$$, we would write $$a \notin A$$.

Notation
Example of a common notation style for the definition of a set: $$A = \{ x \in S \mid P(x) \}$$
 * $$S$$ is a set
 * $$P(x)$$ is a property (The elements of $$S$$ may or may not satisfy this property)
 * Set $$A$$ can be defined by writing:

This would read as "the set of all $$x$$ in $$S$$, such that $$P$$ of $$x$$."

Elements
There are two ways that we could show which elements are members of a set: by listing all the elements, or by specifying a rule which leaves no room for misinterpretation. In both ways we will use curly braces to enclose the elements of a set. Say we have a set $$A$$ that contains all the positive integers that are smaller than ten. In this case we would write $$A = \{1,2,3,4,5,6,7,8,9\}$$. We could also use a rule to show the elements of this set, as in $$A = \{a:\text{ }a\text{ positive integer less than 10}\}$$.

In a set, the order of the elements is irrelevant, as is the possibility of duplicate elements. For example, we write $$X = \{ 1, 2, 3 \}$$ to denote a set $$X$$ containing the numbers 1, 2 and 3. Or,$$X = \{ 1, 2, 3 \} = \{3,2,1 \} = \{1,1,2,3,3\}$$.

Subsets
Formal universal conditional statement: "set A is a subset of a set B" Negation: If and only if: for all $$x$$: If: ($$x$$ is an element of A)        then: ($$x$$ is an element of B) then: set A is a subset of set B Truth Table Example:
 * $$A \subseteq B \Leftrightarrow \forall x$$, if $$x \in A$$, then $$x \in B. $$
 * $$A \nsubseteq B \Leftrightarrow \exists x$$ such that $$x \in A$$ and $$x \notin B. $$

A proper subset of a set is a subset that is not equal to its containing set. Thus

A is a proper subset of B $$\iff$$

Set Identities
Let all sets referred to below be subsets of a universal set U.

(a) A ∪ ∅ = A and (b) A ∩ U = A.

5. Complement Laws:

(a) A ∪ A c = U and (b) A ∩ A c = ∅.

6. Double Complement Law:

(A c ) c = A.

7. Idempotent Laws:

(a) A ∪ A = A and (b) A ∩ A = A.

8. Universal Bound Laws:

(a) A ∪ U = U and

(b) A ∩ ∅ = ∅.

{| class="wikitable" !Identity !For all sets A ! $$A \cup \emptyset = A$$
 * Identity Laws:
 * Identity Laws:

$$A \cup A^c = U$$ $$A \cap A^c =\emptyset$$ $$A \cap A = A$$
 * Complement Laws:
 * Complement Laws:
 * Double Complement Law:
 * $$\left( A^c \right)^c = A$$
 * Idempotent Laws:
 * $$A \cup A =A$$
 * Idempotent Laws:
 * $$A \cup A =A$$
 * Idempotent Laws:
 * $$A \cup A =A$$
 * Universal Bound Laws:
 * }
 * Universal Bound Laws:
 * }
 * }
 * }

2. Associative Laws: For all sets A, B, and C,

(a) (A ∪ B) ∪ C = A ∪ (B ∪ C) and

(b) (A ∩ B) ∩ C = A ∩ (B ∩ C).

3. Distributive Laws: For all sets, A, B, and C,

(a) A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C) and

(b) A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).

For all sets A and B,

De Morgan’s Laws:

(a) (A ∪ B) c = A c ∩ B c and (b) (A ∩ B) c = A c ∪ B c.

Absorption Laws:

(a) A ∪ (A ∩ B) = A and (b) A ∩ (A ∪ B) = A.

Set Difference Law:
 * A − B = A ∩ B c.

Complements of U and ∅:
 * $$U^c= \empty$$
 * $$\empty^c = U$$

Cardinality
The cardinality of a set is the number of elements in the set. The cardinality of a set $$ A $$ is denoted $$ |A| $$.

Types of Sets by Cardinality
A set can be classified as finite, countable, or uncountable.
 * Finite Sets are sets that have finitely many elements, $$ A = \{1,2,3\} $$ is a finite set of cardinality 3. More formally, a set $$A$$ is finite if a bijection exists between $$A$$ and a set $$\{1,\ldots,n\} $$ for some natural number $$n$$. $$n$$ is the said set's cardinality.
 * Countable Sets are sets that have as many elements as the set of natural numbers. As since $$ |\mathbb{Q}| = |\mathbb{N}|$$, the set of rational numbers is countable.
 * Uncountable Sets are sets that have more elements than the set of natural numbers. As since $$ |\mathbb{R}| > |\mathbb{N}|$$, the set of real numbers is uncountable.

Common Sets of Numbers

 * $$\mathbb{N}$$ is the set of Naturals
 * $$\mathbb{Z}$$ is the set of Integers
 * $$\mathbb{Q}$$ is the set of Rationals
 * $$\mathbb{R}$$ is the set of Reals
 * $$\mathbb{C}$$ is the set of Complex Numbers

Comparison of Sets
2 sets $$A$$ and $$B$$ have the same cardinality (i.e. $$|A| = |B|$$), if there exists a bijection from $$A$$ to $$B$$. In the case of $$ |\mathbb{Q}| = |\mathbb{N}|$$, they are the same cardinality as there exists a bijection from $$\mathbb{Q}$$ to $$\mathbb{N}$$.

Partitions of Sets
In many applications of set theory, sets are divided up into non-overlapping (or disjoint) pieces. Such a division is called a partition.

Two sets are called disjoint if, and only if, they have no elements in common."A and B are disjoint ⇔ A ∩ B = ∅."

Sets $$A_{1}, A_{2} , A_{3}, \ldots$$ are mutually disjoint (or pairwise disjoint or nonoverlapping)

if, and only if, no two sets $$A_{i}$$ and $$ A_{j}$$ with distinct subscripts have any elements in

common. More precisely, for all $$i, j = 1,2,3,\ldots $$

$$A_{i} \cap A_{j} = \emptyset$$ whenever $$i \not= j$$.

Power Sets
The power set of a set A is all possible subsets of A, including A itself and the empty set. Which can be represented: $$P_{(A)} = \{ \emptyset, \{ 1 \}, \{ 2 \}, \ldots \}$$

For the set $$ A = \{1,2,3\}$$

$$P_{(A)} = \{\emptyset,\{1\},\{2\},\{3\},\{1,2\},\{1,3\},\{2,3\},\{1,2,3\}\}$$

Wikiversity

 * Introduction_to_set_theory/Lecture_1

Wikipedia

 * Set theory category