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If f maps A onto B and g maps B onto C, is g ∘ f onto C? Small arrow diagrams suggest the composite keeps injections, surjections and bijections, and this video turns the surjection case into a proof by starting at an arbitrary c and pulling it back through g and then through f.
Key concepts covered:
– Injections and surjections recalled for composite functions
– Arrow-diagram tests: composing two injections, two surjections, two bijections
– A composite of surjections that is onto but not one-to-one
– The theorem: composites of injections, surjections and bijections
– Forward-backward method: write out what g ∘ f onto C means
– Choose an arbitrary c, then use each surjection once to find b and then x
– Why the search runs backward while the composite runs forward
– Existence without formulas: the argument covers infinite sets too
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SOURCE MATERIALS
The source materials for this video are from
Attribution: Adapted (narrated and animated) from Mathematical Reasoning: Writing and Proof (PreTeXt Edition, based on Version 3) by Ted Sundstrom, Grand Valley State University ( licensed CC BY-NC-SA 3.0 ( Changes were made: the text was rewritten for narration and the activities worked through. This adaptation is shared under the same license.
License: This video is licensed by Ludium under CC BY-NC-SA 4.0 (
Narration: Built with Fish Audio — Drama 3 open-weights text-to-speech (
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