let $f:S\rightarrow T $ for non-empty sets $S$ and $T$ and let $C$ be a partition of $S$. Define a relation ~ on the set $T$ such that,
$t_1$ ~ $t_2$ if $\exists_{A\in C}$ : $f^{-1}( \{t_1,t_2 \} )$ $\subset$ A.
(a) Need ∼ be reflexive? What if f is injective?
(b) Need ∼ be symmetric? What if f is bijective?
(c) Need ∼ be transitive? What if f is surjective
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Hint:
If $f$ is not onto (surjective), then this relation on $T$ need not be reflexive. For example: Let $S=\{1,2,3\}$ with $C=\{\{1\}, \{2,3\}\}$ and let $T=\{a,b,c,d\}$. Define $f$ as $$f=\{(1,a), (2,b),(3,b)\}.$$ Then $c \not\sim c$ because $f^{-1}(\{c\}) = \phi.$ By the definition of partition, $\phi \not\subset C$.
Even if $f$ is injective, say $$f=\{(1,a), (2,b),(3,c)\}.$$ then too $d \not\sim d$.
I hope this will help you approach the problem. In case you are unable to then please feel free to ask for more explanation.