Homework Answers
Let f : A rightarrow D and g : B rightarrow C be functions. For each...
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?
how do u do 6? F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f...
how do u do 6?
F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answers. (a) f(x) = **!, for all real numbers x #0 (6) f(x) = x, for all real numbers x (c) f(x) = 3x=!, for all real numbers x 70 (d) f(x) = **, for all real numbers x 1 (e) f(x) = for all real...
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f ...
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)? If yes then give a proof and derive a formula for f. If no then explain why not
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)?...
7. (10 points) Let Sym(Z) = \f : Z Z : f bijective) be the set...
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. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers)...
1. a) Let A = {2n|n ∈ ℤ} (ie, A is the set of even numbers) and define function f: ℝ → {0,1}, where f(x) = XA(x) That is, f is the characteristic function of set A; it maps elements of the domain that are in set A (ie, those that are even integers) to 1 and all other elements of the domain to 0. By demonstrating a counter-example, show that the function f is not injective (not one-to-one). b)...
3. The identity function on the set X is denoted by ix and is defined by...
3. The identity function on the set X is denoted by ix and is defined by ix(x) = x for all x E X. It is known that f: X Y and g: Y X are functions. (a) Prove that if go f = ix, then f is one to one. (b) Give an example of f and g with gofrix but g is not one to one, (c) Prove that if go f = ix is onto, then g...
Short Answer Prove the statement For any real number x, [[x]]=[x] TTT Arial 3 (12pt) -...
Short Answer Prove the statement For any real number x, [[x]]=[x] TTT Arial 3 (12pt) - T - = - = - S es Path:p Words:0 QUESTION 13 For the functions below, indicate whether the function is: Onto One-to-one Bijection Example one-to-one not the function f is one-to-one f: {0, 1}+{0, 1}. The output of f is obtained by taking the input string and reversing the bits. For example f(011) = 110. For the functions below, indicate whether the function...
1. Let A -(a, b) a, b Q,a b. Prove that A is denumerable. (You may cite any results from the text...
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...
6. Let A and B be some finite sets with N elements. • Prove that any...
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?
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...
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}...
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?
how do u do 6?
F-'(C-D)= F-'(C)-F-'(D). 4. (10 points) In following questions a function f is defined on a set of real numbers. Determine whether or not f is one-to-one and justify your answers. (a) f(x) = **!, for all real numbers x #0 (6) f(x) = x, for all real numbers x (c) f(x) = 3x=!, for all real numbers x 70 (d) f(x) = **, for all real numbers x 1 (e) f(x) = for all real...
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)? If yes then give a proof and derive a formula for f. If no then explain why not
Let the function f R R be given by 1,)- f 1 z-1 Draw the graph of f versus the values of z. Is f a bijection (i.e., one-to-one and onto)?...
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
3. The identity function on the set X is denoted by ix and is defined by ix(x) = x for all x E X. It is known that f: X Y and g: Y X are functions. (a) Prove that if go f = ix, then f is one to one. (b) Give an example of f and g with gofrix but g is not one to one, (c) Prove that if go f = ix is onto, then g...
Short Answer Prove the statement For any real number x, [[x]]=[x] TTT Arial 3 (12pt) - T - = - = - S es Path:p Words:0 QUESTION 13 For the functions below, indicate whether the function is: Onto One-to-one Bijection Example one-to-one not the function f is one-to-one f: {0, 1}+{0, 1}. The output of f is obtained by taking the input string and reversing the bits. For example f(011) = 110. For the functions below, indicate whether the function...
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...
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?
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}...
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