(27) Let f A → B and g : B → C be functions, and assume that their composition gof A→ C is one-to-one. Does this imply that both f and g are one-to-one? Prove your answer. (28) Let f A → B and g B → C be one-to-one functions, and assume that their composition g of A → C is onto. Does this imply that both f and g are onto? Prove your answer.

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: Provide an example of a that is onto. Can this fun Justify your answer.

A:

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: Question 5. Suppose f : A → B and g : X → Y are bijective functions. Let h : A × X → B × Y be a new…

A:

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

A:

Q: Suppose f : A → B is an injective function. Show that there is a function g : B → A such thatg ◦ f =…

A: The objective of this question is to prove that if there is an injective function f from set A to…

Q: Let X = {n EN: 0≤ n ≤ 999} be the set of all numbers with three or fewer digits. Define the function…

A:

Q: If ƒ : R → R and g : R → R are both one-to-one, is f + g one-to-one? If f and g are both onto, is f…

A: As per the guideline I’m supposed to solve only first question from the given questions. And the…

Q: If f : X → Y and g: Y→ Z are functions such that g of is one-to-one, then f is one-to-one.

A:

Q: Suppose f : A →→ B is an injective function. Show that there is a function g : B→ A such that gof=…

A: Let's prove this statement step by step. Want to show that for any injective function f:A→B, there…

Q: 2. Let f A → B be a function. Prove that if g, h : B → A are inverse functions of f, then g(b) =…

A:

Q: 1. Show that the inverse of a function is unique. If a function has an inverse, then it only has…

A: Solution:(1)Let f:A→B be an invertible function, we have toshow that it's inverse is unique.…

Q: Prove: There exist functions f, g, h in F such that g o f = h o f, but g is not equal to h.

A: The objective is to prove there exist functions f, g, h in F such that g o f = h o f, but g is not…

Q: 6. Determine the algebraic representation of f-(x) from the tabular representation of ƒ(x) and…

A: If  x,y is a coordinate of the function f(x), then y,x is the coordinates of f-1(x). The coordinates…

Q: Consider a function f: A→ B. Suppose there exists a function g: B→ A such that the composition go f…

A:

Q: 2.12.2: Composition of functions on integers. Consider three functions f, g, and h, whose domain and…

A: A composite function is defined as a function whose values are found by applying the variable to the…

Q: Let f: X --> Y and g, h: Y --> Z be functions. Prove or disprove:  (a) If g of f is 1-1, then f is…

A: (a) Suppose g o f is 1-1. To prove f is 1-1.

Q: Define a function S: Z+ →Z+ as follows: For each positive integer n, S(n) = the sum of the positive…

A:

Q: 3. Let ƒ : A → B be a function. Prove that if ƒ has an inverse, then fis bijective.

A: This question is related to topic function from real analysis.

Q: The mapping f: R → R, f(x) = x2, which of the following are correct? Of is one-to-one. Of is not a…

A: We need to select the correct statements (options) for the mapping f:ℝ→ℝ, f(x)=x2

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

A:

Q: D. Let function g: R → R be defined by g(x) = 2x + 1 5 cx in its domain. Why does your choice of c…

A: