1. (a) Let A be a set. Prove that if f :A → A satisfies f ◦f = 1A, then P = { { x, f(x) } | x ∈ A }, is a partition of A.
(b) Let A be a set, and let f :A → A. Prove that if there exists a positive integer n such that f^n = 1A, then f is bijective.
(c) How many bijective functions f :{1,2,3,4,5,6,7} → {1,2,3,4,5,6,7} satisfy f = f^(-1)?
2. Let f :N → N be the function for which f(n) = n + 1 for each n ∈ N. Note that f is injective. Find all left inverses of f.
3. (a) Let A and B be sets. Prove that the relation R = { (f, g) ∈ F(A, B) | Ker(f) = Ker(g) } on the set F(A, B) of all functions from A to B is an equivalence relation.
(b) Let A = {1,2,3,4,5} and B = {1,2,3}. For f =
1 2 3 4 5
1 1 3 1 1
what is |[f]R|?. Give one element g ∈ [f]R with g/= f.
(b) Let A be a set, and let f :A → A. Prove that if there exists a positive integer n such that f^n = 1A, then f is bijective.
(c) How many bijective functions f :{1,2,3,4,5,6,7} → {1,2,3,4,5,6,7} satisfy f = f^(-1)?
2. Let f :N → N be the function for which f(n) = n + 1 for each n ∈ N. Note that f is injective. Find all left inverses of f.
3. (a) Let A and B be sets. Prove that the relation R = { (f, g) ∈ F(A, B) | Ker(f) = Ker(g) } on the set F(A, B) of all functions from A to B is an equivalence relation.
(b) Let A = {1,2,3,4,5} and B = {1,2,3}. For f =
1 2 3 4 5
1 1 3 1 1
what is |[f]R|?. Give one element g ∈ [f]R with g/= f.
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