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.