Let $X$ be the set $\lbrace -12, 3, -13 \rbrace$. For the first three parts of this problem you are asked to define a function $f: X \rightarrow X$ so that the relation

$$u \sim w \Leftrightarrow w = f( u )$$
satisfies each of the following conditions.

(a) $\sim$ is reflexive

$x$	$-12$	$3$	$-13$
$f( x )$

(b) $\sim$ is symmetric

$x$	$-12$	$3$	$-13$
$f( x )$

(c) $\sim$ is transitive

$x$	$-12$	$3$	$-13$
$f( x )$

(d) [optional: see your instructor] Let $Y$ be an arbitrary non-empty set. Determine all functions $g: Y \rightarrow Y$ so that the relation

$$a \sim b \Leftrightarrow b = g( a )$$
is
- Reflexive
- Symmetric
- Transitive