Any function induces a surjection by restricting its codomain to its range. This should be obvious. What may or may not be obvious is that any onto function induces a bijection defined on a quotient of its domain. Let f: A → B be an onto function. Let Q be the set of all equivalence classes of domain A under the equivalence relation x ~ y if and only if f(x) = f(y). (Equivalently, Q is the set of all pre-images under f). Prove that ~ is an equivalence relation.
nnbboouuusssccaaallllllll Long time no see
For this proof we will need to show reflexive, symmetric, and transitive. They're all pretty easy, because it's just equality under the function. Reflexive: f(x)=f(x), Symmetric: f(x)=f(y) implies f(y)=f(x), Transitive: f(x)=f(y) and f(y)=f(z) implies f(x)=f(z).
what does ~ mean?
relation?
Tilde is the standard notation for an equivalence relation. You can use it for any relation, but usually it's used specifically for equivalence relations.
thhaank youuuuuuu
Join our real-time social learning platform and learn together with your friends!