Transitive, Reflexive, Symmetric Relations

Question

I've looked all around this stackexchange before asking this question. Just don't want to get penalized for a repeating question in case there is one.

I'm having trouble finding relations that are

1. transitive but not reflexive or symmetric
2. reflexive and transitive but not symmetric (not quite sure that this is possible)
3. symmetric and transitive but not reflexive

To clarify, I'm looking for three different relations.

Thanks in advance!

Answer 1

Let $A = \{1,2,3\}$, and $R$ be a relation on $A$.

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

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

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

Answer 2

HINTS:
$(1)$ A relation $R$ in $\mathbb R$ with $R=\{(a,b): a<b\}$.
$(2)$ A relation $R$ in $\mathbb R$ with $R=\{(a,b): a^3\geq b^3\}$.
$(3)$ Mentioned in @Fib1123's comment. EDIT: If you want another example, take a set $A=\{a,b\} $ where $a,b $ are distinct. Now take a relation $R $ in $A $ with $R=\{(a,a)\} $. Sorry for the wrong example.