File:Axiom of countable choice.svg
Summary
Demonstration of axiom of choice. Each set in the family of sets <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1c1bef8b6851bc4c925f5ad3308ed3ef5c244a" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:20.552ex; height:2.843ex;" alt="{\displaystyle (S_{i})=S_{1},S_{2},S_{3},...}"> (indexed over the natural numbers N) is nonempty and contain a (potentially infinite) number of elements. The axiom of choice allows us to arbitrarily select a single element from each one, forming a corresponding family of elements <img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c91866f7a47d7815db205b8aa2116c62262624f" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:20.171ex; height:2.843ex;" alt="{\displaystyle (x_{i})=x_{1},x_{2},x_{3},...}">. In general both collections may be indexed over any set, not just N.
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| Date/Time | Thumbnail | Dimensions | User | Comment | |
|---|---|---|---|---|---|
| current | 19:17, 15 January 2017 | | 286 × 324 (61 KB) | 127.0.0.1 (talk) | Demonstration of axiom of choice. Each set in the family of sets <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo stretchy="false">(</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>S</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle (S_{i})=S_{1},S_{2},S_{3},...}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4d1c1bef8b6851bc4c925f5ad3308ed3ef5c244a" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:20.552ex; height:2.843ex;" alt="{\displaystyle (S_{i})=S_{1},S_{2},S_{3},...}"></span> (indexed over the natural numbers N) is nonempty and contain a (potentially infinite) number of elements. The axiom of choice allows us to arbitrarily select a single element from each one, forming a corresponding family of elements <span><span class="mwe-math-mathml-inline mwe-math-mathml-a11y mw-math-element" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow class="MJX-TeXAtom-ORD"><mstyle displaystyle="true" scriptlevel="0"><mo stretchy="false">(</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mi>i</mi></mrow></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>2</mn></mrow></msub><mo>,</mo><msub><mi>x</mi><mrow class="MJX-TeXAtom-ORD"><mn>3</mn></mrow></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo></mstyle></mrow><annotation encoding="application/x-tex">{\displaystyle (x_{i})=x_{1},x_{2},x_{3},...}</annotation></semantics></math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8c91866f7a47d7815db205b8aa2116c62262624f" class="mwe-math-fallback-image-inline mw-math-element" aria-hidden="true" style="vertical-align: -0.838ex; width:20.171ex; height:2.843ex;" alt="{\displaystyle (x_{i})=x_{1},x_{2},x_{3},...}"></span>. In general both collections may be indexed over any set, not just N. |
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