Set Theory/Review

From Wikibooks, open books for an open world
< Set Theory
Jump to: navigation, search

Need help creating math symbols?



A \subseteq B

  • \{x \mid x\in A \hbox{  then  } x\in B \}

Subset means for all x, if x is in A then x is also in B.

Proper Subset[edit]

A \subset B

  • \{x \mid x \in A \hbox{ then } x \in B \hbox{ and } A \ne B\}


\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 \}


\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[edit]


  • \hbox{There is a set } A \hbox{ s.t. } \{x \mid x \notin A\}


A - B

  • \{x \mid x \in A \hbox{ and } x \notin B \}



  • \{x \mid x \subseteq A \}

Ordered Pair[edit]

\langle a, b \rangle

  • \{ \{a\}, \{a, b\}\}

Cartesian Product[edit]

A \times B

  • 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 \}


A set of ordered pairs


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


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


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

Equivalence Relations[edit]

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

Partial Ordering[edit]

  • Transitive and,
  • Irreflexive: for all a, <a, a> not in R


Exactly one of the following holds

  • x < y
  • x = y
  • y < x

Proof Strategies[edit]

If, then[edit]

Prove if x then y

Suppose x
so, y

If and only If[edit]

Prove x iff y

suppose x
so, y
suppose y
so, x


Prove x = y

show x subset y
show y subset x


Prove x != y

x = {has p}
y = {has p}
a in x, but a not in y