# Examples of irreflexive and antisymmetric binary relations

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

#### Solution Preview

(a) irreflexive

Recall that a binary relation R on a set S is irreflexive if there is no element "x" of S such that (x, x) is an element of R.

Let S = {a, b}, where "a" and "b" are distinct, and let R be the following binary relation on S:

R = {(a, b), (b, a)}

Then R is irreflexive, because neither (a, a) nor (b, b) is an element of R.

Recall that, for any binary relation R on a set S, R^2 (R squared) is the binary relation

R^2 = {(x, y): x and y are elements of S, and there exists z in S such that (x, z) and (z, y) are elements of R}

For our relation R, note that both (a, b) and (b, a) are in R, so (letting x = a, y ...

#### Solution Summary

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.