Image of Intersection under Mapping/Examples/Square Function

Example of Image of Intersection under Mapping
Let:
 * $S_1 = \set {x \in \Z: x \le 0}$
 * $S_2 = \set {x \in \Z: x \ge 0}$


 * $f: \Z \to \Z: \forall x \in \Z: \map f x = x^2$

We have:
 * $f \sqbrk {S_1} = \set {0, 1, 4, 9, 16, \ldots} = f \sqbrk {S_2}$

Then:
 * $f \sqbrk {S_1} \cap f \sqbrk {S_2} = \set {0, 1, 4, 9, 16, \ldots}$

but:
 * $f \sqbrk {S_1 \cap S_2} = f \sqbrk {\set 0} = \set 0$

As can be seen, the inclusion is proper, that is:
 * $f \sqbrk {S_1 \cap S_2} \ne f \sqbrk {S_1} \cap f \sqbrk {S_2}$

Also see

 * Image of Intersection under Injection: equality holds $f$ is an injection.