# Appendix 11. Set Theory (mathematical logic)

*Mathematical Set Theory* is functional way to study and understand the relations between things or features. It is here presented in a very simple form and only through its main results. The idea is to use this theory as a base for our thinking.

*Image 79. The Sets A and B and intersection A∩B highlighted*

In this example, two groups are presented and they are marked with big letters ‘A’ and ‘B’. There also can be members in the groups, which are marked as small letters. Let Set A be all the *Americans* and let Set B all the *Europeans*.

**Unions**

If we choose to study both Americans and Europeans, we would study both Sets A and B. Here, we would use term Union. Union of Sets A and B is marked A∪B and it means that both Sets are under considered at the same time. Union includes all the members that are a member of A, or B, or both.

**Complements**

If a person is American, but not European, he belongs to A. Also, if a person is European, but not American, she belongs to B. If all the people we study belong to either A or B, and not any of them to both Sets A and B, then A and B are complements. Complements means that the Sets do not have any common members. A complement Set is usually expressed A^c, which means the complement set of A.

**Intersection**

If there would be a person x that is American and European (having two citizenships), then we would say that x belongs to the Intersection of A and B – x belongs to both Sets A and B. The intersection of Sets is usually marked with an arch such as A∩B.

**Empty** **Set**

Then if we choose to study only the member y, who is Asian person, we would know that Sets A and B still exist, but because we only study the member y, we would say that A and B are Empty Sets – they have no members inside. An Empty Set is usually marked with ∅. Thus, if there would be only Americans, then it would be valid to say A≠∅,and B=∅.

**Subsets**

Then if there would be a war, where Europeans would conquer the Americans, Set B would include Set A. Thus it would be valid that every element of A is an element of B. Then we would say that A is a subset of B, or B includes A. It is marked with A ⊂ B. Please notice also that if A is a subset for B and also B is a subset for A, then A must be B. This is one technique of mathematical proving.