Please solve 2 and 3. 2. Let f : [1,00)- [2, oo) be defined by f(z-z...
7. (10 points) Let Sym(Z) = \f : Z Z : f bijective) be the set of bijective functions from Z to Z. (Sym(Z),o) is a group, where o denotes the composition of functions. Let g: Z Z be the function 8(n) = {-1 nodd n+1 neven (a) Prove that g € Sym(Z). (b) Find the order of g. Heat: gog - composition of functions
1. Let f: R\{-1} → R, f() = 2+1 (a) Prove that f is not an increasing function on its domain, but its restrictions to intervals fl-20.-1) and fl(-1,00) are strictly increasing. (b) Find a codomain for fl-1.00) that makes the function bijective. Find the composi- tional inverse of our function. Sketch both our function and its inverse on the same set of axes.
5. Let A = P(R). Define f : R → A by the formula f(x) = {y E RIy2 < x). (a) Find f(2). (b) Is f injective, surjective, both (bijective), or neither? Z given by f(u)n+l, ifn is even n - 3, if n is odd 6. Consider the function f : Z → Z given by f(n) = (a) Is f injective? Prove your answer. (b) Is f surjective? Prove your answer
7. Using the definition of continuity directly to prove that f: (1,00) + R defined by f(3) = and f(1) = 0 is continuous on at 2 = 2 but not continuous at 1.
Let f:R → Z defined by f(x) = 23 – 2. Prove that f is a one-to-one correspondence (i.e., a bijection).
Solve and show work for problem 8
Problem 8. Consider the sequence defined by ao = 1, ai-3, and a',--2an-i-an-2 for n Use the generating function for this sequence to find an explicit (closed) formula for a 2. Problem 1. Let n 2 k. Prove that there are ktS(n, k) surjective functions (n]lk Problem 2. Let n 2 3. Find and prove an explicit formula for the Stirling numbers of the second kind S(n, n-2). Problem 3. Let n 2...
How do I prove this function is not surjective?
3.) Let f: R-R, f(x)-x2+ x+1 and Show that f is not injective and not surjective Justify that g is bijective and find gt. PIR, Show all the wortky) Not Surtechive: fx) RB Surjective: ye(o,oo) hng (g) 8 gon)-es is bijecelive g(x)-ex+s
Abstract Algebra
Answer both parts please.
Exercise 3.6.2 Let F be a field and let F = FU {o0) ( where oo is just a symbol). An F-linear fractional transformation is a function T: given by ar +b T(z) = cr + d ac). Prove that the set where ad-be 0 and T(oo) a/c, while T(-d/c) = o0 (recall that in a field, a/c of all linear fractional transformations M(F) is a subgroup of Sym(F). Further prove that if we...
The graph of f is shown to the right. The function F(z) is defined by F(z) = f f(t) dt for 0 x 4. a) Find F(0) and F(3). 2 b) Find F (1). c) For what value of z does F(z) have its maximum value? What is this maximum value? d) Sketch a possible graph of F. Do not attempt to find a formula for F. (You could, but it is more work than neces- sary.) -1
The graph...
Please answer all!!
17. (a) Let R be the relation on Z be defined by a R b if a² + 1 = 62 + 1 for a, b e Z. Show that R is an equivalence relation. (b) Find these equivalence classes: [0], [2], and [7]. 8. Let A, B, C and D be sets. Prove that (A x B) U (C x D) C (AUC) Ⓡ (BUD).