Week 6
... the first fundamental theorem says that F 0 (t) = f (t). Before stating and proving the result, we need to introduce an important concept called connectivity. We need to discuss when two points x, y in our open set D can be joined by a path α : [a, b] → D with α(a) = x and α(b) = y. Even for n = 1 t ...
... the first fundamental theorem says that F 0 (t) = f (t). Before stating and proving the result, we need to introduce an important concept called connectivity. We need to discuss when two points x, y in our open set D can be joined by a path α : [a, b] → D with α(a) = x and α(b) = y. Even for n = 1 t ...
Functions Definition of Function Terminology Addition and
... Let f be a function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: We must show that f-1(S) ⊆ f-1(S) and that f-1(S) ⊆ f-1(S) . Let x ∈ f-1(S). Then x∈A and f(x) ∉ S. Since f(x) ∉ S, x ∉ f-1(S). Therefore x ∈ f-1(S). Now let x ∈ f-1(S). Then x ∉ f-1(S) which implies that f(x) ...
... Let f be a function from A to B. Let S be a subset of B. Show that f-1(S) = f-1(S) Proof: We must show that f-1(S) ⊆ f-1(S) and that f-1(S) ⊆ f-1(S) . Let x ∈ f-1(S). Then x∈A and f(x) ∉ S. Since f(x) ∉ S, x ∉ f-1(S). Therefore x ∈ f-1(S). Now let x ∈ f-1(S). Then x ∉ f-1(S) which implies that f(x) ...
