Recipe
Classify a relation as reflexive, irreflexive, or neither
Use when: you're given a relation R on a set A and asked whether every, no, or only some elements relate to themselves
Goal: the correct label -- reflexive, irreflexive, or neither -- with the specific pair that proves it
Taught in Reflexive and irreflexive
Quick recipe
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1
Write out the full set A the relation is defined on; this is the exact list of self-pairs (a,a) that need checking.
Example
A=\{1,2,3\} means the self-pairs to check are (1,1), (2,2), (3,3).
Why: reflexivity and irreflexivity are properties of R relative to a specific set A, so the pairs to check change if A changes.
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2
Go through the list from the first step one element at a time, marking whether (a,a) is in R or not.
Example
For R=\{(1,1),(1,2),(2,3)\} on \{1,2,3\}: (1,1) is in R, but (2,2) and (3,3) are not.
Why: a single missing or extra self-loop already decides the classification, so every element needs checking, not just a few.
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3
All self-pairs present means reflexive; all self-pairs absent means irreflexive; a mix means neither.
Example
Continuing above: (1,1) is present, but (2,2) and (3,3) are missing, so R is neither reflexive nor irreflexive.
- every (a,a) is in R — R is reflexive.
- no (a,a) is in R — R is irreflexive.
- some (a,a) are in R and some are not — R is neither.
Why: reflexive and irreflexive are each an all-or-nothing condition across every element, not a majority rule.
Common mistakes
- Concluding reflexive from a rule "looking like" it should hold, without checking whether it actually holds on every element of the specific set A.
- Stopping after checking only some elements and generalizing to the whole set.
- Treating "not reflexive" as the same as "irreflexive" -- a relation missing just one self-loop is not reflexive, but needs every self-loop missing to be irreflexive.
- Forgetting that changing the set A can flip the answer even though the relation's rule stays the same.