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

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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:

\(\ds S_1\) \(=\) \(\ds \set {\tuple {m, 1}: m \in \N}\)
\(\ds S_2\) \(=\) \(\ds \set {\tuple {0, 2 n}: n \in \N}\)


First note that we have:

\(\ds S_1 \cap S_1\) \(=\) \(\ds \set {\tuple {m, 1}: m \in \N} \cap \set {\tuple {0, 2 n}: n \in \N}\)
\(\ds \) \(=\) \(\ds \O\) as $1$ is not an integer of the form $2 n$


Then:

\(\ds \pr_1 \sqbrk {S_1}\) \(=\) \(\ds \set {m: m \in \N}\)
\(\ds \) \(=\) \(\ds \N\)
and:
\(\ds \pr_1 \sqbrk {S_2}\) \(=\) \(\ds \set 0\)
\(\ds \) \(=\) \(\ds 0\)
\(\ds \leadsto \ \ \) \(\ds \pr_1 \sqbrk {S_1} \cap \pr_1 \sqbrk {S_2}\) \(=\) \(\ds \N \cap 0\)
\(\ds \) \(=\) \(\ds 0\)


while:

\(\ds \pr_1 \sqbrk {S_1 \cap S_2}\) \(=\) \(\ds \pr_1 \sqbrk {\O}\)
\(\ds \) \(=\) \(\ds \O\)


As can be seen, the inclusion is proper, that is:

$\pr_1 \sqbrk {S_1 \cap S_2} \ne \pr_1 \sqbrk {S_1} \cap \pr_1 \sqbrk {S_2}$


Sources

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