This paper is about basic set theory and cardinalities for infinite sets. One of the results are that the line R and the plane R2 contains exactly the same number of points. Because of that the set theory is described with a formal language this the paper has an appendix about formal languages..

Ternary codes for unequal error protection is a part of a communication system where different parts of the information to be transmitted can receive different amount of error protection. There are a lot of applications where the channel conditions fluctuate so that reliable or acceptable communication can not be guaranteed. Regardless whether the fluctuations of the channel are intentional or not the idea is to let the coderate decrease so that the error correcting capability can increase for the part of the information to be more protected. By using the support, the nonzero positions of the ternary code, a new binary code can be extracted with properties different from the ternary code in the sense of Cardinality and minimum Hamming distance. When the channel conditions are good the receiver decodes the message using ternary decisions and receives all information included in the codeword.