Problem 15.19. Let f : A -> B be a function and CCA (a) Prove that if f is one-to-one, then flc is one-to-one (b) Pr...
6. Let A and B be some finite sets with N elements. • Prove that any onto function : A B is an one-to-one function. • Prove that any one-to-one function /: A B is an onto function. • How many different one-to-one functions f: A+B are there?
Consider the function ?:ℤ×ℤ×ℤ→ℤ, defined by ?(?,?,?)=?2?−?3. a) Is ?f a one-to-one function? Prove or disprove. b) Is ?f an onto function? Prove or disprove.
1. (a) (6 points) Let f : A + B and g:B + C be two functions. Suppose that the composition of functions go f is a bijection. Prove that the function f : A + B must be one-to-one and that the function g:B + C must be onto. (b) (4 points) Give an example of a pair of functions, f and g, such that the composition gof is a bijection, but f is not onto and g is...
(c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a < x < b.) (c) Let f :la,b- R be an integrable function. Prove that lim . (Your argument should include why faf makes sense for a
4. Let A be a non-empty set and f: A- A be a function. (a) Prove that f has a left inverse in FA if and only if f is injective (one-to-one) (b) Prove that, if f is injective but not surjective (which means that the set A is infinite), then f has at least two different left inverses.
hello sir, solve both questions Problem 5: Let f : A → B be a function, and let X-A and Y-B. Show that X S(x)) Problem 6: Recall that BA denot es the set of all functions A the function f : P(A) → {0,1}A by B. Fix a set A and defi ne f (X)Xx (the charact erist ic function), VX EP(A) Prove that f is a bijection
1. Let A -(a, b) a, b Q,a b. Prove that A is denumerable. (You may cite any results from the text.) 2. Let SeRnE N) and define f:N-+S by n)- n + *. Since, by definition, S-f(N), it follows that f is onto (a) Show that f is one-to-one (b) Is S denumerable? Explain 3. Either prove or disprove each of the following. (You may cite any results from the text or other results from this assignment.) (a) If...
11. (a) Let F be a field. Prove FixF Rİr (b) Let R be a commutative ring with identity. Prove that one can have R. 11. (a) Let F be a field. Prove FixF Rİr (b) Let R be a commutative ring with identity. Prove that one can have R.
Let P be the power set of {a, b, c}. A function f: P , the set of integers, follows: For A in P, f(A) = the number of elements in A. 1. Is f one-to-one? Explain. 2. Is f onto? Explain.
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.