# Relations: reflexive, antisymmetric, transitive

For the set A = {a, b, c}, let R be the relation on A which is defined by the following 3 by 3 matrix M_R:

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Row 1: 1 0 1

Row 2: 1 1 0

Row 3: 0 1 1

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Which of the properties (reflexive, antisymmetric, transitive) are satisfied by R?

Begin your discussion by defining each property in general, and then determine whether R satisfies that property.

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#### Solution Preview

We use the positions of the 1's in the matrix M_R to determine R (i.e., to determine the set of ordered pairs of elements of the set {a, b, c} which belong to R):

From the first row of M_R (which indicates the elements (a, x) in R, for x in A), we see that (a, a) and (a, c) are in R (but not (a, b)), because the first and third elements of the first row are the locations of the 1's.

From the second row of M_R (which indicates the elements (b, x) in R, for x in A), we see that (b, a) and (b, b) are in R (but not (b, c)), because the first and second elements of the second row are the locations of ...

#### Solution Summary

Definitions are given of the following properties of a binary relation: reflexive, antisymmetric, and transitive. A detailed determination of which of these are properties of the given relation is presented.

Reflexive, Antisymmetric and Transitive Properties

Please see the attached file for the fully formatted problems.

Let A = {1, 2, 3, 4, 5, 6,12} and define the relation R on A by m R n iff

m|n.

Write the definitions of the properties, reflexive, antisymmetric and transitive and the use

the definitions to determine whether each property holds for this relation.

(a) Is this relation a partial ordering relation? Why? If so, draw its Hasse

diagram.

(b)Write the (boolean, that is the yes/no) matrix of this relation.

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