The continuous image of a connected set is connected. That is, if X is connected and f : X–→ Y is continuous, thenf [X] is connected.

Let f A B be a function. Let X be a subset of A. (a) Verify that X ≤ f−¹(ƒ(X)). (b) Give an example of sets X CA and B, and a function f : A → B such that f−¹(f(X)) £ X. (c) Prove that if f is injective then X = f−¹(ƒ(X)).

Let A and B be sets. Let f : A → B be a function. Let S ⊆ A be a subset. Prove that graph(f|S) = graph(f) ∩ (S × B).

Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if X1 ⊆ X2, then f(X1) ⊆ f(X2). b) Prove that if Y1 ⊆ Y2, then f -1 (Y1) ⊆ f -1(Y2)

Determine the types of the following partial di erential equations(a) 4uxx4uxy+ uyy= 0(b) uxx+ 2uxy+ 2uyy= 0(c) uxx+ 3uxy+ 2uyy= 0