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
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections: Exercise $1$