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Relations on Sets

Published online by Cambridge University Press:  23 January 2015

Des MacHale  and
Peter MacHale
Des MacHale
Affiliation:
School of Mathematical Sciences, University College Cork, Cork, Ireland
Peter MacHale
Affiliation:
Cork Constraint Computation Centre, University College Cork, Cork, Irelande-mails:, d.machale@ucc.ie , p.machale@4c.ucc.ie
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Extract

This paper is dedicated to the memory of George Boole FRS (1815-1864), first Professor of Mathematics at University College Cork.

A relation on a non-empty set K is any subset W of the cartesian product K × K. If (x, x) ∈ W for every x ∈ K, W is said to be reflexive on K. If for x, y, z ∈ K, (x, y) ∈ W implies that (y, x) ∈ W, W is said to be symmetric on K. If for x, y, z ∈ K (x, y) ∈ W and (y, z) ∈ W together imply that (x, z) ∈ W, W is said to be transitive on K. Finally, if the relation W is reflexive, symmetric and transitive on K, W is said to be an equivalence relation on K. We contend that equivalence relations are of central importance in many areas of mathematics. Many texts on algebra (for example Topics in Algebra [1]) contain as an exercise the following fallacious argument which the reader is challenged to refute.

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 539 , July 2013 , pp. 224 - 233
Copyright
Copyright © The Mathematical Association 2013

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References

1. Herstein, I. N., Topics in algebra, Xerox College Publishing (1975).Google Scholar
2. Jordan, C. R., Transitivity of relations, Math. Gaz. 57 (March 1973) pp.64–65.Google Scholar
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