Image of Intersection under Mapping/Examples/First Projection on Subsets of Cartesian Natural Number Space

Example of Image of Intersection under Mapping
Let $\pr_1: \N \times \N \to \N$ denote the first projection from the cartesian space $\N \times \N$ of the natural numbers.

Let:

First note that we have:

Then:

while:

As can be seen, the inclusion is proper, that is:
 * $\pr_1 \sqbrk {S_1 \cap S_2} \ne \pr_1 \sqbrk {S_1} \cap \pr_1 \sqbrk {S_2}$