Image of Intersection under Mapping/Examples/Square Function
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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 if and only if $f$ is an injection.
Sources
- 1971: Allan Clark: Elements of Abstract Algebra ... (previous) ... (next): Chapter $1$: Mappings: $\S 12 \alpha$
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 21$: The image of a subset of the domain; surjections: Postscript to $\S 21.4 \ \text{(i)}$