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 a ∈ A. Another element, let’s say m is not within A, therefore we can write m ∉ A.

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 S ⊆ B.

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

The universal Set U includes all objects and additionally itself. So all sets are also subsets of the universal set B ⊆ U and S ⊆ U.

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