Mathematical basics – Set theory

Set

A Set is a collection of objects that are distinct. Assume the following Set A has the elements a, b, c and d, so we can say A = {a,b,c,d}.

In the example above, the element a is within Set A. You can write aA. Another element, let’s say m is not within A, therefore we can write mA.

Two sets have the same mightiness, if they can be imaged bijectively on each other, i.e. there is a one-to-one relationship between their elements.

Bijection:

  • Complete pair formation between the elements of the definition set and target set
  • Bijections thus treat their definition range and their value range symmetrically
  • definition set and target set have the same mightiness

Subsets

If all elements of a Set S are also within another Set B, we call S a subset of B, or SB.

  • all elements of A are also elements of B
  • expressed as SB
  • also S is contained in B
  • same BS, means B is a superset of S
  • all sets are subsets of themselves: SS and BB

The universal Set U includes all objects and additionally itself. So all sets are also subsets of the universal set BU and SU.

The empty set, expressed as ∅, is a subset of all other sets, like as example B and B

Power sets

The Power Set of a Set contains all subsets of the set and additionally also the empty set. The following example defines a set S with elements {1,2}. The Power Set P(S) contains als subsets and the empty set.

  • Power set of finite set contains 2^n elements

Isomorphism

  • bijective
  • also Homeomorphism

Homeomorphism

  • bijective
  • structure-preserving
  • weaker than Isomorphism

Diverse

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