# 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