Need help understanding this question, its related to one problem. Thanks!
The following functions f, g, h are defined:
f:ℝ -> ℝ, ∀x ∈ ℝ, f(x) = x^2.
g: ℕ -> ℕ, ∀x ∈ ℕ, g(x) = x^2.
h: A -> B, ∀x ∈ A, h(x) = x^2. A = {0,1,2,3,4} and B = {0,1,4,9,16}.
Qn 1: Let P:C -> D be a function, and let U ⊆ C. We denote P_u to mean the restriction of P to U. That is P_u : U -> D, ∀x ∈ U, P_u(x) = P(x).
Which of the following statement is true?
A. ∃U ⊆ ℝ such that f_u = g.
B. ∃U ⊆ ℕ such that g_u = f.
C. ∃U ⊆ A such that h_u = g.
D. ∃U ⊆ A such that h_u = f.
Answer: A (Would like to check what it means by f_u = g? is it f(u) gives the domain of g which is ANY natural number? What does the answer actually mean?)
Qn 2: Which of the following statements is true?
A: h ∘ h is a relation
B: g^-1 is a function
C: g ∘ g is one-one
Answer: C
(Why A is wrong? Since an example will be 2->4->2. How can I know when a composition is a relation?)
(Why g^-1 isnt a function? Since the range and domain are Natural numbers should it be a function? g is also a bijection to begin with isn't it?)