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Sets ​

sets are the basic building blocks of mathematics.

Definition: ​

A set is a well defined unordered collection of distinct elements.

Examples: ​

  • {1,3,6} a set of numbers

  • {Jan,Pier,Tjoris,Corneel} a set of guys with beards

  • {{1,3,6},{Jan,Pier,Tjoris,Corneel}} a nested set

  • {1,2,3,…} a infinite set

  • {1,3,6,6}={1,3,6}Distinct property

  • {1,3,6}={1,6,3}Unordered property

  • collection of good soccer clubs≠setNot well defined

Notation: ​

  • List of all elements:
    {1,2,3,4,5,6,7,8,9}

  • Pattern:
    {1,2,...,9}

  • Properties:
    {n∣n is whole number with 1≤n≤9}

Set Relations ​

A set consist of elements.

Being an element or not relates an object to a set.

  • 1 is an element of the set {1,3,6}

    • 1∈{1,3,6}

  • 2 is not an element of the set {1,3,6}

    • 2∉{1,3,6}

Subsets: ​

Definition Subset: ​

A⊆B when every element a ∈A is also an element of B. (if a ∈A, then a∈B)

  • {1,3} is a subset of {1,3,6}

    • {1,3}⊆{1,3,6}

  • {1,3} is a proper subset of {1,3,6}

    • {1,3}⊂{1,3,6}

{1,3,6} is a subset of itself but not a proper subset of itself.

Definition Proper Subset: ​

A⊂B and A≠B

Definition Set Equality: ​

A=B when A⊆B and B⊆A

Example: ​

Is {a,b} a subset of {b,{a,b}} ?

  • {a,b}⊈{a,{a,b}}
  • {a,b}∈{a,{a,b}}

Proposition vs. Predicate: ​

Proposition: statement for which you can determine if it's truth value (True or False).

Predicate: a statement for which the truth value (T/F) cannot be determined. (missing value for a variable)

If you add a Domain to a Predicate it becomes a Proposition because you can now determine the truth value (T/F).

Symbols: ​

MeaningSymbol
Element ofa∈A
Subset ofA⊆B
Proper subsetA⊊B
Empty set∅
UnionA∪B
IntersectionA∩B
DifferenceA/B
ComplementA―

Sources ​

  • Harry Aarts, Ed Brinksma, Jan Willem Polderman, Gerhard Post, Marc Uetz, Marjan van der Velde (2018) Introduction to Mathematics
  • Micro-lectures wk1