Let F(R, R) be the set of all functions from R to R. For any f, g € F(R, R), we define fog € F(R, R) by setting (fog)(a) = f(g(a)) for all a € R. (a) Prove or disprove that o is associative. (b) Prove or disprove that is commutative.

Q: Let f: A→B be a function. Prove or disprove the equivalence: (a) f is injective (b) If C is any set…

A:

Q: Does there exist a set of all functions? Suppose such a set exists and call it F. Define a…

A:

Q: 5. Let X be the set of odd integers and Y = {a € Z | a is even and a ≤ 20}, and consider the…

A: (a) Let, f(x) = f(y)          => x(9 – x) = y(9 – y)          => 9x - 9y - x2 + y2 = 0…

Q: Let A = {1, 3, 5, 6, 7, 9, 10, 12} and B = {1, 2, 3, 5, 7, 8, 9, 11, 12}. Consider the following…

A: Given a function f: A -> B , we have to determine whether it is injection, surjection,…

Q: A = {1,2,3}, B = {4,5,6}, C = {7,8,9,10} define the functions f :A→C and g:B→C by f(1) = 7, f(2) =…

A:

Q: . Consider the functions f : A → B and g : B → C. a) If f and g are both injective, show that g o f.…

A:

Q: (d) Given a function F : P({a, b, c}) → Z is defined by F(A) = |A| for all A E P({a,b, c}). i. Is Fa…

A: For the given function F(A)=|A| F will be one one if A=B implies F(A)=F(B) F will be onto if for a…

Q: 1.3.2. Let f: X → Y be a function, let B be a subset of Y, and let {B;}ie1 be a family of subsets of…

A: Let f:X→Y be a function, let B be a subset of Y and Bii∈I be a family of subsets of Y.  To prove…

Q: (c) h: Z → Z. h(x) = x³ (d) f: Z - Z, f(x) E - 4 (e) f: Z- Z, f(x) = 5x - 4

A:

Q: 6 language of a function f being continous at a point zo € dom(ƒ). Let f(x) = 2x² +1. Using the e- &…

A:

Q: A function f : Z Z is defined by f(n) = 3n + 1. Let E be the set of even %3D integers. Determine the…

A: When a function is linear, it is one-one and onto function. We know that a bijective function is…

Q: 3. Let A be a finite set. In this exertise functions f, g will be of type g: A → A and ƒ: A → A. (a)…

A:

Q: Suppose f : A → A is a function and A has 7 elements. Prove each direction of the following…

A:

Q: Suppose that f : A → B is a function, viewed as a relation (set of ordered pairs). Let f* denote the…

A:

Q: Let F be the set of all functions of the form f : N {0, 1}. Also, let G be the set of all functions…

A: We have to find all possible options that are correct for the given question.

Q: How many functions f: {1,2,3,4} {1,2,3,4} are there satisfying: a. f(1) = 1 or f(2)= 2 (or both)? b.…

A: Solution :-

Q: 5. Let f: Z → Z be defined by f(n) = n² + 1. (a). Is f a one-to-one function? Explain. (b). Is f an…

A: Given- The function is f(n)=n2+1 and f:ℤ→ℤ. Definition- Let a function from X to Y, One-to-one: If…

Q: A = {2, 3, 4, 5} and B et f : A → B be a func ow. Which of the following

A: A function f: A → B is said to be an ONTO function if, The range that exists for f is the set B…

Q: How many finite functions exist of the form f: {1, 2, 3,...,m} {1, 2, 3,..., n}? How many of them…

A:

Q: Given that f is a function from A onto B, that g :B →C, and that the function g • f is one-one,…

A: In the given question we have to find f and g are one - one function.

Q: Let (A,) be a poset. Define a function f: A → P(A) by f(x) = {a € A: a <x}. What can you say about x…

A: Here given that be a poset.Define function is  by Noted that here  is power set of  .Here we have to…

Q: 2. Suppose f : X →Y and g : Y → Z are functions, and g of :X → Z is injective. (a) Show that f is…

A:

Q: Prove that fo f = Idom f iff f-¹ is a function from ran f to dom f.

A: Prove That : f-1of=Idom iff f-1is a function from ran f to dom f.

Q: 1. Consider the functions f : A → B and g : B → C. (a) If f and g are both surjective, show that so…

A:

Q: 3. Let A = {1, 2, 3, 4}. Let F be the set of all functions from A to A. (a) How many pairs (f, g) E…

A:

Q: (1) Let f be a function from A to B, {An} a sequence of subsets of A, and {B} a sequence of subsets…

A:

Q: THEOREM Let f be a function from A into B. If A, and A2 are subsets of A, then (b) f(A, N A2) C f(A)…

A: We solve this by drawing related picture.

Q: Let f, g be entire functions such that f(z)| ≤ lg(z)| for all z E C. Show that there exists a cEC…

A: Given that f and g are entire functions such that |f(z)|≤|g(z)| for all z∈ℂ. We have to show that…

Q: Let R be a relation on the set of all functions from Z to Z defined by: R = {(f,9)|f(1) = g(3) or…

A:

Q: For a set A, let {0,1} A be the set of all functions from A to the two point set {0,1}. Prove that…

A:

Q: How many functions f: {1,2,3,4} → a. ƒ(1) = 1 or ƒ(2) = 2 (or both)? → {1, 2, 3, 4} are there…

A: Since your question carry more than three subparts , so as per Bartleby guidelines we are answering…

Q: Let X be the set {5, –6, 2}. For the first three parts of this problem you are asked to define a…

A: Consider the set 5,-6,2. The function is f:X→X and the relation is u~w⇔w=fu.   a) The relation is…

Q: Determine whether the function f from { a, b, c, d} to { a, b, c, d} is injective (one - to - one),…

A:

Q: 7. For each of the following, give an example of functions f: A → B and g: B C that satisfy the…

A:

Q: Let A = {1,2,3, ...,8} and consider the function ƒ: P(A) → N given by ƒ (B) = |B|. Prove or disprove…

A: Given A={1,2,3,....,8}  and the function f: Ƥ(A) → ℕ here   Ƥ(A) is power set of A and  'f' map from…