1) In F[x] where F is a field, if f is irreducible and whenever f=gh for some g,h in F[x], then g is a unit or _____. 2) In F[x], f is a unit in F[x] if and only if f is a/an ______. 3) Let f be in ℤ[x]. We say that f is irreducible over ℤ if and only if f is ____ and f is irreducible over ℚ. 4) R must be at least a/an _____ so that deg(fg) = deg(f)+deg(g) for f,g in R[x]. 5) We say that r is a _____ iff f(r)=0

Q: following must be true?

A: As per the definition of function if f(a) = f(b) then a must be equal to b.   In given question f(a)…

Q: Show that if both f: A ---> B and g: B ----> C are bijective, then g 0 f: A ---> C is bijective.

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Q: Give an example of a function that is one-to-one but not onto. Give an example of a function that is…

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Q: Give an example of a function from the set of all integers to the set of all negative even integers…

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Q: Suppose that f is a function from A to B, where A and B are finite sets with |A| = |B|. Show that f…

A: Given that f is a function from A to B, where A and B are finite sets with A=B. We have to show that…

Q: Let x, y ∈ F, where F is an ordered field. Suppose 0 < x < y. Show that x2 < y2.

A: multiplying by x and y in x&lt;y respectively and then comparing both result.....

Q: Let C([0, 1]) denote the set of all continuous functions f : [0, 1] → R, and let Qn (0, 1] =…

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Q: Let ƒ : R → R be a function. Give precise mathematical formulations of the following statements…

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Q: Determine whether the function f is one-to-one, onto, or both. Prove your answers. The domain and…

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Q: Write out all functions f: {1,2} {a, b, c, d) (in two-line notation). How many functions are there?…

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Q: Give an example of a function from the set of all integers to the set of all positive even integers…

A: Given, Domain = Set of all integers. Co-domain = Set of all positive even integers. The set of all…

Q: Let A = {-4, 0, 4} and B = {t, u, v, w}. Define a function F: A → B by the following arrow diagram:…

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Q: assume that f is an even function and g is an odd function. Both f and g are defined on the real…

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Q: Consider functions f and g such that composite gof is defined and is one-one. Are f and g both…

A: Given f and g such that composite gof is defined and is one-one.

Q: Let A = {x ∈ R| x≠0}, and then define f: A->R by:           f(x) = 6x+10/2x         is f…

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Q: Which one of the following statements concerning polynomials is true? Select one: a. Every…

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Q: c) Describe the set of all 7. For a function f:A → B, the statement "f is surjective" is defined by…

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Q: Let ℝ+ denote the set of positive real numbers. Define a relation E on ℝ+ as follows: a E b  if and…

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Q: Let f: A -> B and g: B -> C be two functions. Prove: if f and g are surjective, then gof is…

A: Given: Let  f: A→B and g: B→C be two functions.  If f and g are Surjective(onto), To Prove:  g∘f:…

Q: Define f : {1,2, 3, 4} → {1,2,3, 4, 5} by ƒ(x) = x + 1. Give a left inverse g : {1,2, 3, 4, 5} →…

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Q: Let f: A --> B and g: B --> C. Suppose that g of f is bijective. Show that g is onto and that f is…

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Q: Prove the following statements. (a) If f and g are bijective then f-¹ og¯¹ ℃ (gof)¯¹. (b) If f is…

A: This is a question of mapping and set theory.

Q: Prove the statement. Let f : A → B . If X ⊆ A, Y ⊆ B , and f-1[Y] = {x ∈ A|f(x) ∈ Y}, then f[X] ∩ Y…

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Q: Suppose f: R→ R and g: R → R are functions, and the composite function fog is a bijection. Which of…

A: introduction If a product f: A B meets for both invertible (yet another product) and done by…

Q: Let f = (1, 6) (2, 3, 5, 4). Compute each of the following powers of f.   f-2

A: We have given the permutation :  f=(1,6)(2,3,5,4) We have to compute : f-2

Q: Prove the following statements. (a) If f and g are bijective then f-1 og ¹ C (gof)-¹. (b) If f is…

A: This is a problem of function.

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Q: Suppose that f is holomorphic in an open set 2. Prove that in any one of the following cases: 2 3. f…

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Q: A is a non-empty set. Show that the power set of A is in one to one correspondence with the set of…

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Q: Let X = {1,2,3,4} and Y = {5,6,7}. State whether a function f : X → Y can be one-to-one or not…

A: By using definition of one-time and onto function we solve the given problem :

Q: Let X= {a,b,c,d) and Y = {e, f,g}. Define functions F and G by the arrow diagrams below. Select the…

A: From the given diagram, we see,         For any element in Y, there is a pre-image in X under F, but…

Q: Determine if the following statement is true or not. Let the domain be the set of all integers Z. O…

A: Statements and conclusions