For each of the functions f_1, f_2, f_3, f_4 defined below, determine if it is injective, surjective or neither. (a) Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and let X = {B ∈ P(A) : |B| > 1} . Define f_1 : X → A by f_1(B) = smallest element of B for B ∈ X. (Here P(A) is the power set of A and |B| denotes the cardinality of the set B.)
For each of the functions f_1, f_2, f_3, f_4 defined below, determine if it is injective, surjective or neither.
(a) Let A = {1, 2, 3, 4, 5, 6, 7, 8, 9} and let X = {B ∈ P(A) : |B| > 1} . Define f_1 : X → A by
f_1(B) = smallest element of B
for B ∈ X. (Here P(A) is the power set of A and |B| denotes the cardinality of the set B.)
(b) Let M = {0, 1, 2, 3, 4, 5, 6} and define f_2 : M → M by
f_2(x) = {2x if x < 4
= {2x − 7 if x ≥ 4.
(c) Let p be a permutation of the set {1, 2, 3, 4}. Let f_3 = p ◦ p be the composition of p with itself. Note: the domain and codomain of f_3 are the same as for p.
(d) Define f_4 : R → R by x^3 − 6x^2 + 3x
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