Set Theory Relations

Question

Given set $A={1, 2, 3}$ consider the following relation on $A$

$R=\{(1, 1), (2, 1), (3, 3), (3, 2)\}$

Which one of the following statements are true

1. $R$ is antisymmetric and transitive
2. $R$ is antisymmetric but not transitive
3. $R$ is not transitive and not antisymmetric
4. $R$ is reflexive and antisymmetric

I have 4 as the answer, because:

• $R$ is reflexive: $(1, 1), (3, 3)$
• $R$ is antisymmetric: $(2, 1), (3, 2)$ are in $R$ but not $(1, 2), (2, 3)$

But there may also be a case for transitivity because of $(2, 1), (3, 2)$

I'm at a bit of a loss really

Answer 1

The relation is not reflexive because $(2,2)\notin R$. Remember that the definition of reflexivity is that $(a,a)\in R$ for all $a\in A$.

Also, the relation is not transitive, because $(3,2)\in R$ and $(2,1)\in R$, but $(3,1)\notin R$.

Finally, a relation is antisymmetric if the conditions $(a,b)\in R$ and $(b,a)\in R$ imply $a=b$. Is this true in the $R$ you've given?