Relations / Relation Properties
Least You Need to Know: Relation Properties
A relation can be **reflexive**, **symmetric**, **antisymmetric**, or **transitive**. The skill is recognizing what each word really asks you to check.
The least you need to know
- Reflexive means every element is related to itself.
- Symmetric means whenever a is related to b, then b is related to a.
- Antisymmetric does not mean 'not symmetric'.
- Transitive means chains can be shortened: if aRb and bRc, then aRc.
Key notation
R
a relation
aRb
a is related to b
(a,a)
self-pair
Tiny worked example
- Relation on {1,2,3}: R = {(1,1),(2,2),(3,3),(1,2),(2,1)}.
- It is reflexive because all self-pairs are present.
- It is symmetric because (1,2) and (2,1) appear together.
- It is not antisymmetric because 1R2 and 2R1 but 1 ≠ 2.
Common mistakes
- Students often think antisymmetric means the opposite of symmetric.
- Students often test only one example and stop.
- Students often forget to check all self-pairs for reflexive.
How to recognize this kind of problem
- Look for ordered pairs and property words like reflexive or transitive.
- For property questions, build a checklist and test one property at a time.
- For transitive, inspect two-step chains.