Give an example of a binary relation R such that R is irreflexive but R^2 (R squared) is not irreflexive, and give an example of a binary relation R such that R is antisymmetric but R^2 is not antisymmetric.
For each of the following properties, find a binary relation R such that R has that property but R^2 (R squared) does not:
(a) irreflexive
(b) antisymmetric
The definitions of the two given types of binary relations (irreflexive relation and antisymmetric relation), and the definition of the square of a binary relation, are reviewed. An example of a binary relation R such that R is irreflexive but R^2 is not irreflexive is provided, including a detailed explanation of why R is irreflexive but R^2 is not irreflexive. Also, two different examples of a binary relation R such that R is antisymmetric but R^2 is not antisymmetric are given, including a detailed explanation (for each example) of why R is antisymmetric but R^2 is not antisymmetric.
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