Recipe
Classify a relation as symmetric or antisymmetric
Use when: you're given a relation R on a set A and asked whether every arrow has a return arrow, no distinct pair does, or both
Goal: the correct symmetric and antisymmetric labels, with the specific pair that proves or disproves each one
Taught in Symmetric and antisymmetric
Quick recipe
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1
Write out R as a full list of pairs (a,b); both checks below scan this same list.
Example
R = \{(1,2),(2,1),(2,3)\} on \{1,2,3\} is the starting list for both checks.
Why: both symmetric and antisymmetric are defined pair by pair, so an incomplete list of R produces an incomplete check.
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2
For each pair (a,b) in R, look for (b,a) also in R. One missing reverse breaks symmetric for the whole relation.
Example
(2,3) is in R, but (3,2) is not, so symmetric already fails.
Why: symmetric requires (b,a) to follow from (a,b) for every single pair, not most of them.
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3
Look for two different elements a and b with both (a,b) and (b,a) in R. Finding even one such pair breaks antisymmetric.
Example
(1,2) and (2,1) are both in R, and 1 \ne 2, so antisymmetric fails too.
- no distinct pair has its reverse present — antisymmetric holds.
- some pair of distinct elements has both (a,b) and (b,a) present — antisymmetric fails.
Why: antisymmetric only forbids a round trip between two different elements; a self-pair (a,a) never violates it.
Common mistakes
- Treating "not symmetric" as automatically meaning "antisymmetric" -- checking one property says nothing about the other on its own.
- Flagging a self-pair (a,a) as breaking antisymmetric, when only a round trip between two distinct elements counts.
- Stopping the symmetric check after finding one matching reverse pair, instead of checking every pair in R.
- Missing that a relation with no off-diagonal pairs at all is both symmetric and antisymmetric simultaneously.