Concise Guide to Formal Methods pp 65-92 | Cite as

# Sets, Relations and Functions

## Abstract

This chapter provides an introduction to fundamental building blocks in mathematics such as sets, relations and functions. Sets are collections of well-defined objects; relations indicate relationships between members of two sets *A* and *B*; and functions are a special type of relation where there is exactly (or at most) one relationship for each element *a* ϵ *A* with an element in *B*. A set is a collection of well-defined objects that contains no duplicates. A binary relation *R* (*A*, *B*) where *A* and *B* are sets is a subset of the Cartesian product (*A* × *B*) of *A* and *B*. A total function *f*: *A* → *B* is a special relation such that for each element *a* ϵ A there is exactly one element *b* ϵ B. This is written as *f*(*a*) = *b*. A partial function differs from a total function in that the function may be undefined for one or more values of *A*.

### References

- 1.D. Hoffman, D.L. Parnas, in
*Software Fundamentals*, ed. by D. Weiss. Collected Papers by D.L. Parnas (Addison Wesley, Reading, 21)Google Scholar - 2.E.F. Codd, A relational model of data for large shared data banks. Commun. ACM
**13**(6), 377–387 (1970)Google Scholar - 3.C.J. Date, in
*An Introduction to Database Systems*. 3rd edn. The Systems Programming Series (1981)Google Scholar - 4.G. O’ Regan,
*Introduction to the History of Computing*(Springer, Switzerland, 2016a)Google Scholar - 5.G. O’ Regan,
*Guide to Discrete Mathematics*(Springer, Switzerland, 2016b)Google Scholar - 6.D. Turner,
*Miranda*, in Proceedings IFIP Conference, Nancy France, Springer LNCS (201) (September 1985)Google Scholar