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Composition/Double names/Permutation of names/Exercise

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Let be the set of all female compound first names (names of the form , where are simple first names and every combination is allowed). We consider the composition

which sends a pair of double first names to the double name .

  1. What is the value of under this composition?
  2. Is this composition commutative?
  3. Is this composition associative?
  4. Does this composition have a neutral element?
  5. Is this composition surjective?
  6. Is this composition injective?