Bijections

• If g∘f Is Bijective, Is f Injective and g Surjective?
  If f:A→B and g:B→C, bijectivity of g∘f forces f to be injective and g to be surjective. See the proof and examples.
• Why the Map f(x) = ax Is Bijective in a Group
  For a fixed element a in a group G, left multiplication f(x)=ax is bijective because its inverse is multiplication by a⁻¹.
• Are the Positive Multiples of 17 and 11 in Bijection?
  The positive multiples of 17 and 11 are in bijection: match 17k with 11k for each positive integer k, and verify the map is one-to-one and onto.
• Show That the Identity Function on Any Set Is a Bijection
  For any set A, the identity map sends each element to itself, making it injective and surjective—and therefore bijective, even when A is empty.