Section 2: Problem 1 Solution »

Section 2: Functions

The way Munkres defines functions.
  • A rule of assignment is a subset of the cartesian product such that if and then .
    • The domain of is the set .
    • The image set of is the set .
  • A function is a rule of assignment together with a set that contains the image set of as a subset.
    • The range of is the set .
  • is used to indicate that is the domain of and is the range of .
Let , , .
  1. preserves inclusions and unions: , , , .
    1. The last two are equalities if is injective, i.e. in this case it preserves all four operations.
  2. preserves inclusions, unions, intersections and differences: , , , .
  3. ( if is injective), ( if is surjective).
  4. If is injective then in injective and is injective on .
    1. If has a left inverse then is injective.
  5. If is surjective then is surjective, moreover, it is surjective on .
    1. If has a right inverse then is surjective.
  6. In general, a function may have more than one left inverse or more than one right inverse, but if it has both a left inverse and a right inverse then is bijective and both inverse functions are equal, therefore, the inverse is unique.