Set Theory/Review

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= Definitions =

Subset
$$A \subseteq B$$ Subset means for all x, if x is in A then x is also in B.
 * $$\{x \mid x\in A \hbox{ then } x\in B \}$$

Proper Subset
$$A \subset B$$
 * $$\{x \mid x \in A \hbox{ then } x \in B \hbox{ and } A \ne B\}$$

Union
$$\bigcup A$$
 * $$\{x \mid x \in \bigcup A \hbox{ iff } y \in A \hbox{ s.t. } x \in y \}$$

$$A \cup B$$
 * $$\{x \mid x \in A \hbox{ or } x \in B \} $$

Intersection
$$\bigcap A$$
 * $$\{x \mid \hbox{for all } a \in A, x \in a\} $$

$$A \cap B$$
 * $$\{x \mid x \in A \hbox{ and } x \in B\} $$

Empty Set
$$\empty$$
 * $$\hbox{There is a set } A \hbox{ s.t. } \{x \mid x \notin A\} $$

Minus
$$A - B$$
 * $$\{x \mid x \in A \hbox{ and } x \notin B \} $$

Powerset
$$\mathcal{P}(A)$$
 * $$\{x \mid x \subseteq A \} $$

Ordered Pair
$$\langle a, b \rangle$$
 * $$\{ \{a\}, \{a, b\}\} $$

Cartesian Product
$$A \times B$$ or
 * $$A \times B = \{ x \mid x = \langle a, b \rangle \hbox{ for some } a \in A \hbox{ and some } b \in B \} $$
 * $$ \{ \langle a, b \rangle \mid a \in A \hbox{ and } b \in B \} $$

Relation
A set of ordered pairs

Domain
$$\{x \mid \hbox{ for some } y, \langle x, y \rangle \in R\}$$

Range
$$\{y \mid \hbox{ for some } x, \langle x, y \rangle \in R\} $$

Field
$$\hbox{dom(} R\hbox{)} \cup \hbox{ran(}R\hbox{)} $$

Equivalence Relations

 * Reflexive: A binary relation R on A is reflexive iff for all a in A,  in R
 * Symmetric: A rel R is symmetric iff for all a, b if  in R then  R
 * Transitive: A relation R is transitive iff for all a, b, and c if  in R and  in R then  in R

Partial Ordering

 * Transitive and,
 * Irreflexive: for all a,  not in R

Trichotomy
Exactly one of the following holds
 * x < y
 * x = y
 * y < x

= Proof Strategies =

If, then
Prove if x then y
 * Suppose x
 * so, y
 * so, y
 * so, y

If and only If
Prove x iff y
 * suppose x
 * so, y
 * suppose y
 * so, x
 * suppose y
 * so, x
 * so, x
 * so, x
 * so, x

Equality
Prove x = y
 * show x subset y
 * and
 * show y subset x

Non-Equality
Prove x != y
 * x = {has p}
 * y = {has p}
 * a in x, but a not in y