Set Theory/Review

< Set Theory

DefinitionsEdit

SubsetEdit

 

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

Proper SubsetEdit

 

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UnionEdit

 

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IntersectionEdit

 

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Empty SetEdit

 

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MinusEdit

 

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PowersetEdit

 

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Ordered PairEdit

 

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Cartesian ProductEdit

 

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or

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RelationEdit

A set of ordered pairs

DomainEdit

 

RangeEdit

 

FieldEdit

 

Equivalence RelationsEdit

  • 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 OrderingEdit

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

TrichotomyEdit

Exactly one of the following holds

  • x < y
  • x = y
  • y < x

Proof StrategiesEdit

If, thenEdit

Prove if x then y

Suppose x
...
...
so, y

If and only IfEdit

Prove x iff y

suppose x
...
...
so, y
suppose y
...
...
so, x

EqualityEdit

Prove x = y

show x subset y
and
show y subset x

Non-EqualityEdit

Prove x != y

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