is some function and is injection
If is surjective then we want to prove that must be surjective
That is, we want to show that for every , there exists such that
As is a function, we have exists
And so we have is a surjective function means such that
That is,
But we already have
So that via injectivitiy of
And so such that
Which means must be surjective
3. Let f be a continuous function on [a, b] with f(a)0< f(b). (a) The proof of Theorem 7-1 showed that there is a smallest x in [a, bl with f(x)0. If there is more than one x in [a, b] with f(x)0, is there necessarily a second smallest? Show that there is a largest x in [a, b] with f(x) -0. (Try to give an easy proof by considering a new function closely related to f.) b) The proof...
Formal proof and state which proof style you use Let a function where f:Z5 → Z5 defined by f(x) = x3 (mod5). a. Is f an injection? Prove or provide a counter example. b. Is fa surjection? Prove or provide a counter example. c. Find the inverse relation of f. Verify that it is the inverse, as we have done in class. d. Is the inverse of f a function? Explain why it is or is not a function.
A. (Leftovers from the Proof of the Pigeonhole Principle). As before, let A and B be finite sets with A! 〉 BI 〉 0 and let f : A → B be any function Given a A. let C-A-Va) and let D-B-{ f(a)} PaRT A1. Define g: C -> D by f(x)-g(x). Briefly, if g is not injective, then explain why f is not injective either. Let j : B → { 1, 2, 3, . . . , BI}...
7. Let f be an entire function. Suppose there exists € >0 such that f(2) > € for every 2 E C. Show that f is constant. (Hint: Apply Liouville's theorem to the function g(2) = 1/f().)
7. For a function f:A B, the statement "f is surjective" is defined by the quantifier expression Consider the statement "(E uEA)(V v E B)(f(u) variables has been reversed. v)", in which the order of the quantified a) In intuitive terms, what does this new statement mean? b) Are there functions for which this new statement is true? If so, are they all surjective? c) Are there surjective functions f for which this new statement is false?
Let X be a set with an equivalence relation ∼. Let f : X/ ∼→ Y be a function with domain as the quotient set X/ ∼ and codomain as some set Y . We define a function ˜f, called the lift of f, as follows: ˜f : X → Y, x 7→ f([x]). We define a function Φ : F(X/ ∼, Y ) → F(X, Y ), f 7→ ˜f. (1) Is Φ injective? Give a proof or a...
(b) ONLY! Though you can use the result from (a) without proof (a) Let F(x) = x + x2 + x3 +... and let G(x) = x - x2 + x3 – x4.... Show that for k > 1 and n>k, (4")F(x)* = (n = 1) and if n < k then [x"]F(x)k = 0. (b) Show that G(F(x)) = x.
Let f : A rightarrow D and g : B rightarrow C be functions. For each part, if the answer is yes, then prove it, otherwise give a counterexample. Suppose f is one-to-one (injective) and g is onto (surjective). Is go f one-to-one (injective)? Suppose f is one-to-one (injective) and g is onto (surjective). Is g f onto (surjective)? Suppose g is one-to one. Is g one-to-one? Suppose g f onto. Is g onto?
Let f(x) = { 80 -5 if < 10 - 7+ + b if : > 10 If f(x) is a function which is continuous everywhere, then we must have b = Let f(x) = 82 - 5 if x < 10 1 - 7x +b if x > 10 If f(x) is a function which is continuous everywhere, then we must have b= -6 2-5 - 2x + b if - 1 Let f(x) if 2 - 1 There...