Set Theory/Review

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

Need help creating math symbols?

Definitions[edit]

Subset[edit]

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

Proper Subset[edit]

Union[edit]


Intersection[edit]

Empty Set[edit]

Minus[edit]

Powerset[edit]

Ordered Pair[edit]

Cartesian Product[edit]

or

Relation[edit]

A set of ordered pairs

Domain[edit]

Range[edit]

Field[edit]

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

Trichotomy[edit]

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

Equality[edit]

Prove x = y

show x subset y
and
show y subset x

Non-Equality[edit]

Prove x != y

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