Algebraic Generalization of Venn Diagram

It is easy to deal with a Venn Diagram for 1 ≤ n ≤ 3 sets. When n gets larger, the picture becomes more complicated, that's why we thought at the following codification. That’s why we propose an easy and systematic algebraic way of dealing with the representation of intersections and unions of many sets. Introduction. Let's first consider 1 ≤ n ≤ 9, and the sets S1, S2, ..., Sn. Then one gets 2-1 disjoint parts resulted from the intersections of these n sets. Each part is encoded with decimal positive integers specifying only the sets it belongs to. Thus: part 1 means the part that belongs to S1 (set 1) only, part 2 means the part that belongs to S2 only, ..., part n means the part that belongs to set Sn only. Similarly, part 12 means that part which belongs to S1 and S2 only, i.e. to S1∩S2 only. Also, for example part 1237 means the part that belongs to the sets S1, S2, S3, and S7 only, i.e. to the intersection S1∩S2∩S3∩S7 only. And so on. This will help to the construction of a base formed by all these disjoint parts, and implementation in a computer program of each set from the power set P(S1∪ S2∪...∪ Sn) using a binary number. The sets S1, S2, ..., Sn, are intersected in all possible ways in a Venn diagram. Let 1 ≤ k ≤ n be an integer. Let’s denote by: i1i2...ik the Venn diagram region/part that belongs to the sets Si1 and Si2 and ... and Sik only, for all k and all n. The part which is outside of all sets (i.e. the complement of the union of all sets) is noted by 0 (zero). Each Venn diagram will have 2 disjoint parts, and each such disjoint part (except the above part 0) will be formed by combinations of k numbers from the numbers: 1, 2, 3, ..., n. Example. Let see an example for n = 3, and the sets S1, S2, and S3. 1 It has been called the Smarandache’s Codification (see [4] and [3]).