Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $A = \{a, b, c\}$ and $R = \{(a, c), (b, b), (c, a)\}$ be a relation on $A$.

Determine whether $R$ is reflexive, symmetric, transitive and anti-symmetric, or not.

share|cite|improve this question
All four are very straightforward; can you at least determine whether $R$ is reflexive? – Brian M. Scott Dec 2 '12 at 3:20
Are you clear about what it means for $R$ to be reflexive, symmetric, transitive, and/or anti-symmetric? – amWhy Dec 2 '12 at 3:25
up vote 5 down vote accepted

Let $A = \{a, b, c\}$ and $R = \{(a, c), (b, b), (c, a)\}$ be a relation on $A$.

We need to have that for all $x \in A$, $(x, x) \in R$.

  • Is this true for $a \in A?\quad$ So...

We need to have that for all $x, y \in A$, if $(x, y) \in A$ then $(y,x)\in A$.

  • Hint: there is only one pair of values to be concerned about: $(a, c) \in R$. If $(c, a)\in R$, then the relation is symmetric.

We need to have that for all $x, y, z \in A$, if $(x, y)$ and $(y, z)$ are in $R$, then $(x, z)$ is in $R$.

  • Note that $(a, c), (c, a) \in R,$ but $(a, a) \notin R.\quad$ So...

We need to have that for all $x, y \in A$, if $(x, y), (y, x) \in R$, then $x = y$.

  • We can see that $(a, c), (c, a) \in R$, but $a \neq c$. So $R$ is not antisymmetric, since it violates the definition of antisymmetry.
share|cite|improve this answer
Thanks a lot. Really helpful . – thejimzee Dec 2 '12 at 4:03
My pleasure, thejimzee! – amWhy Dec 2 '12 at 4:08
Yes i do know what it means. e.g. R is not reflexive if there is an element "a" in A such that (a,a)∉R. That is some element "a" of A is not related to itself. For Symmetric: R is not symmetric if there are elements a and b in A such that {a,b}∈R but (b,a)∉R For Transitive: R is not Transitive if there are elements a, b, c in A such that if(a,b) ∈R and (b,c)∈R but (a,c)∉R please correct me if i am wrong . – thejimzee Dec 2 '12 at 4:13
You've got it! =) – amWhy Dec 2 '12 at 4:19
once again thanks for your time. – thejimzee Dec 2 '12 at 4:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.