Prove that cardinality is an equivalence relation. Prove for all
properties (refextivity, transitivity and symmetry). Please do this
problem and explain every step. The less confusing notation the
better, thanks!
Consider the equivalence relation on Z defined by the
prescription that all positive numbers are equivalent, all negative
numbers are equivalent, and 0 is only equivalent to itself. Let f ∶
Z → {a, b} be the function that maps all negative numbers to a and
all non-negative numbers to b. Does there exist a function F ∶ X/∼→
{a, b} such that f = F ○ π? If so, describe it
.
An automorphism of a group G is an isomorphism from G to G. The
set of all automorphisms of G forms a group Aut(G), where the group
multiplication is the composition of automorphisms. The group
Aut(G) is called the automorphism group of group G.
(a) Show that Aut(Z) ≃ Z2. (Hint: consider generators of Z.)
(b) Show that Aut(Z2 × Z2) ≃ S3.
(c) Prove that if Aut(G) is cyclic then G is abelian.