Set Theory/Review
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Definitions[edit | edit source]
Subset[edit | edit source]
Subset means for all x, if x is in A then x is also in B.
Proper Subset[edit | edit source]
Union[edit | edit source]
Intersection[edit | edit source]
Empty Set[edit | edit source]
Minus[edit | edit source]
Powerset[edit | edit source]
Ordered Pair[edit | edit source]
Cartesian Product[edit | edit source]
or
Relation[edit | edit source]
A set of ordered pairs
Domain[edit | edit source]
Range[edit | edit source]
Field[edit | edit source]
Equivalence Relations[edit | edit source]
- 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 | edit source]
- Transitive and,
- Irreflexive: for all a, <a, a> not in R
Trichotomy[edit | edit source]
Exactly one of the following holds
- x < y
- x = y
- y < x
Proof Strategies[edit | edit source]
If, then[edit | edit source]
Prove if x then y
- Suppose x
- ...
- ...
- so, y
If and only If[edit | edit source]
Prove x iff y
- suppose x
- ...
- ...
- so, y
- suppose y
- ...
- ...
- so, x
Equality[edit | edit source]
Prove x = y
- show x subset y
- and
- show y subset x
Non-Equality[edit | edit source]
Prove x != y
- x = {has p}
- y = {has p}
- a in x, but a not in y