What is the difference between symmetric and antisymmetric

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Connect and share knowledge within a single location that is structured and easy to search. So I'm having a hard time grasping how a relation can be both antisymmetric and symmetric, or neither. The properties of symmetry, antisymmetry, and reflexivity have very simple interpretations in these terms:.

A loop is an edge from some vertex to itself. To make a relation that is neither symmetric nor antisymmetric, just find a digraph that has both a one-way street and a two-way street, like this one:. Look at your second example. Is the relation symmetric? Is the relation antisymmetric? For this reason, you might say the relation is vacuously antisymmetric. The argument for its symmetry is similar.

To violate symmetry or antisymmetry, all you need is a single example of its failure, which Gerry Myerson points out in his answer. Sign up to join this community. So in matrix representation of the asymmetric relation, diagonal is all 0s. And in digraph representation , there are no self-loops.

If you make a set of all type 2 people and they may be Sane or Insane people. Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Is an anti-symmetric and asymmetric relation the same? Are irreflexive and anti reflexive the same? Ask Question. Asked 7 years, 6 months ago. Active 9 months ago. Viewed 64k times. I hope I made the question clear.

Thank you. Add a comment. Active Oldest Votes. An example of symmetric relation : " An example of asymmetric relation : " Every asymmetric relation is also an antisymmetric relation. This is not the same as the formula in the definition. Examples about reflexive and irreflexive or anti-reflexive : the relation " Linear Algebra. Math Practice Questions. Table of Contents. Save Article. Improve Article.

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