Let X and Y be sets, A is a subset of X.
Define functions f: A --> Y and F: X --> Y.
Show that F is an extension of f if and only if
the graph of f is a subset of the graph of F.
extension of function
Let f : X → Y and suppose that B is an arbitrary subset of Y . Show that f(inverse) f^-1 [Y \ B] = X \ f^-1 [B].
3) Let X, Y be vector fields. For all functions f, define the commutator X, Y]0=X(Y()-Y(X(f). Show that X, Y=Z is a vector field, by verifying that it satisfies sum rule and product rule: Z(f+g)-Z(+Z(g) Zfg)-fZ(g)+gZ(). Extra credit: write /X, YJ in local coordinates.
If A and B are sets and f : A → B, then for any subset S of A we define f(S) {be B : b-f(a) for some a ε S). Similarly, for any subset T of B we define the pre-image of T as Note that f (T) is well defined even if f does not have an inverse! Now let fRR be defined as f(x) 2. Let Si denote the closed interval [-2,1], that is all TE R...
Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y . Prove that for every subset A ⊂ X: (a) (10 points) A ⊂ F^(−1) (F(A)). (b) (10 points) F ^(−1) (F(A)) ⊂ A
4. Let X and Y be any sets and let F be any one-to-one (injective) function from X to Y. Prove that for every subset A CX: (a) (10 points) AC F-(F(A)). (b) (10 points) F-1(F(A)) C A.
12. Let g(x), h(y) and p(z) be functions and define f(x, y, z) = g(x)h(y)p(2). Let R= = {(x, y, z) E R3: a < x <b,c sy <d, eszsf} where a, b, c, d, e and f are constants. Prove the following result SS1, 5100,2)AV = L*()dx ["Mwdy ['Plzdz.
9. Let X and Y be metric spaces, and let D be a dense subset of X. (For the definition of "dense, see Problem 4 at the end of Section 3.5.) (a) Let f : X → Y and g : X → Y be continuous functions. Suppose that f(d)gld) for all d E D. Prove that f and g are the same function.
Let X, Y be two nonempty sets and let f : X → Y. For a, b X we write a ~ b iff f(a) = f(b). Prove that~is an equivalence relation on X Write lely for the equivalence class of x e X with respect to “~" Express [ely in terms of the function f: Irl, = {re x : f(z') a: b: ?? J. (I d o not want to see ..|x ' = {x"e X : r,...
(1) Let X and Y be sets. Let f be a function from X to Y, (a) IF BEY, recall that F-'(B) = {xeX \flyeBX(y,x) ef-)}. Prove that f'(B)={xeX | fk)e B}. (hint: Reprember that even though t is a thought is a function, the relation f may well not be itself a function.) Al b) Let {B; \je J} be an inbred family of subsets of Y. Prove that of "b) = f'(21B;).
Problem 2. Let f be a self-map on a set X. For x,y e X define x ~ y if and only if f"(x) = f(y) for some integers n, m > 0. Show that ~ is an equivalence relation.