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The Lumenary
الصفوف 6–8Category Theoryمُخطَّط له

Function machines in sequence

Feeding the output of one function into another, the identity function that returns its input unchanged, inverse functions that undo, and tables and arrow diagrams for chained functions.

math.cat.chaining-functions

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FunctionsOne thing after another

عدد المتطلبات السابقة الإلزامية (حتى الأساس): 9

ستتعلّم

  • Compute the result of applying one function and then another, using tables, arrow diagrams and formulas.
  • Recognise the identity function on a set and explain why chaining it with any function changes nothing.
  • Find inverses of simple bijections, such as x ↦ x + 3 or x ↦ 2x on ℚ, and check that they undo the original.

تعريف

  • Applying f : A → B and then g : B → C sends each a in A to g(f(a)) in C.
  • The identity function on A sends each element of A to itself.
  • A function g undoes f if g(f(a)) = a for every a in the domain of f.

مبرهنة

  • Chaining is possible only when every output of the first function is an allowed input of the second.
  • Chaining with an identity function, before or after, gives back the original function.
  • If f and g can both be undone, then 'f, then g' is undone by 'undo g, then undo f'.

مفاهيم خاطئة شائعة

  • Writing the chain 'first f, then g' as f(g(x)).
  • Thinking every function can be undone; x ↦ x² on ℤ cannot.
  • Ignoring whether the outputs of the first function lie in the domain of the second.

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Diagrams where paths agree

اكتشافات اختيارية