Define  F : ℤ ✕ ℤ → ℤ ✕ ℤ  as follows: For every ordered pair  (a, b)  of integers,  F(a, b) = (2a + 1, 3b − 2). Find the following. (a) F(5, 5) =             (b) F(3, 1) =                (c) F(4, 3) =                (d) F(1, 6) =

Q: 4.) Prove that (F-G)(v)=F(v)-G(v) for all F, GEg (V,W)

A: L(V,W) , a set of linear transformation from V(F) to W(F) , is a vector space over the field F with…

Q: Prove that f(x) = x 3 - 12x is differentiable. Compute f'(x) and find f '(-3), f '(0), f' (2), and…

A: Since, we know  that a polynomial function is always differentiable.⇒f(x)=x3-12x-(i) is always…

Q: 4. If f:(X,r)(Y.A) and g:(Y.A) (Z,r)are closed functions, prove that g of :(X.r)(Z.r) is closed…

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Q: Let F(x, y, z) = vy-V %3D (a) Evaluate F(7, 9, 1) (b) Find the domain of

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Q: Let f: R - R be defined by f(x) = |x+2, r20 interval ]1, 3[ under f) is (a) ]3, 5[U] – 1, 1[ (b) (0.…

A: given that,  f(x)=x-2, x<0x+2, x≥0

Q: Q3/ 1- Let f: Zn → Z10 define by f(x) = 2x+2. Is fhomomorphism or not, explain your answer.

A: Given :-  let f : ℤn → ℤ10 define by f x = 2x + 2.  To Determine :-  Is f homomorphism or not.

Q: Find a codomain B such that ƒ: A → B, ƒ(x) = e²´ is surjective. Show that g: B → A, g(x) = Vln x is…

A: As you asked only subpart (C) and (D) So, I answered here only subpart (C) and (D).

Q: 2. Determine whether f is a function from Z to R if a) f(n) =±n. b) f(n) = Vn²+1. c) f(n) = 1/(n² –…

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Q: Let f(x,y) = x - 2xy – y“. Compute f(3,0) and f(3, – 7).

A: To compute f(3,-7) Given:- f(x,y)=x2-2xy-y2

Q: Define F: R x R → R × R as follows: F(x, y) = (3y – 1,1 – a) for all (x, y) in R × R. (a) F(0,0) = ?…

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Q: Consider the functionf defined by f(x,y) = k. Use the definition to show that for fixed real numbers…

A: We are given that f(x, y) = k and a, b, c, and d are fixed real numbers with a<b and c<d.…

Q: Give an example where | f(x)dg(x) exist even through f is unbounded on [a, b].

A:

Q: Consider the function d: R² × R² → R defined by min{|y2 - x₂, 1} if x₁ = y₁ otherwise, d(x, y) :=…

A: A metric on a set X is defined as a function d: X×X→R which satisfies the following conditions:…

Q: Let f: R (-1, 1) be a function defined by Prove that (a) f is injective; (b) f is surjective. f(x) =…

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Q: (b) Give an example of a bounded function f on [a,b] for which f2 e R[a,b), but f is not in R[a,b).

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Q: Suppose that f'(x) > M for all x E [a, b]. Prove that f(b) > f(a) + M(b

A: Since  f'(x)≥M therefore the function is differentiable and hence continuous let f is continuous on…

Q: 3. Let f: A B and g B → A be functions such that gof = i4. Prove that if g is injective, then f is…

A:

Q: Given a function f: A → B and subsets X,Y C B, prove that f"(XNY) = f(X)nf-(Y).

A: He we will prove the given result.

Q: Prove that the function f: R -→ R defined by f(x)= x³ is bijective.

A: If a function is both one-one and onto, then the function is bijective.

Q: Let f(z) be the principal branch of (z)^⅓, find f(-i)

A: What is Principal Branch: A function that chooses one branch of a multi-valued function is known as…

Q: Let f(x, y) = xy - 4x on

A:

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: Find and sketch the domain of f(r, y) In(r - y) 6. -

A: Given:      f(x,y)=lnx-y9-x2-y2·x2+y2-4    Interval notation : -3,0∪0,3   Set builder notation:…

Q: Let f(x) = x3 + 6 ∈ Z7[x]. Write f(x) as a product of irreduciblepolynomials over Z7.

A:

Q: Consider the function f : R³ → R defined as V(x1, x2, x3) E R³ f(x1, x2, X3) = X1X2+ X1X3 + X2X3 a.…

A: We are given the function f: ℝ3→ℝ defined as ∀x1, x2, x3∈ℝ3    fx1, x2, x3=x1x2+x1x3+x2x3. For (a),…

Q: Let f(æ) = In(z') f'(2) =

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Q: Let f:Z → Z be defined by f (x) = 2x + 1 where x € Z. Is f a one to one correspondence (bijection)?

A: Now let

Q: Define a relation T from R to R as follows.for all real numbers X and Y, (X,Y)€T means that Y2-X2=1…

A:

Q: 5. (a) Consider the function f : C → R, f(2) = (z + z)². Write down the %3D domain, codomain and…

A: The domain is C, the set of all complex numbers. The codomain is R, the set of all real numbers.

Q: 1. Show that the following maping f are linear: f : R² → R² defined by f(x, y) = (x+ y,x).

A:

Q: (a) Prove the following relation: В (x + 1,у): B(x, y) x +y %3D

A: We have give Beta function relation, we have to show left hand side is equal to right hand side.

Q: Define F: Z × Z→Z×Z as follows: For all ordered pairs (a, b) of integers, F(a, b) = (2a + 1, 3b –…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: Let f : Z → Z be relation defined by f(x) = ||æ| – 5|+ x for all æ E Z. i) f is a function ii) f is…

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Q: ww Prove that F+0=F for every Fɛg (V,W)

A: We can find the answer as below.

Q: Find a function f: Q → Q that is both surjective and injective but not the identity function.

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Q: Prove that the function f: R W {2} R {5} defined by - 5x + 1 f(x) = #-2 is bijective.

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Q: Let f be a function from A to B . Define a relation R by (x,y) ∈ R If and only if  f(x) =f(y) .Prove…

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Q: Define f: [0,2]-R by f(x) = x²+2x-1 , then the sup f- inf f is : [0,2] [0,2] .a 3 .b 8 .C 8, .d 1

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Q: Describe the domain of the function f(x, y) = In(4- x- y) %3D O {x,y):y<-x+4} O (x, y):x20, y 2 0} O…

A: The answer is given in step-2.

Q: Let f : R → R be the function defined by 1 f(x) 2* a) Is ƒ is one-to-one? If yes, then explain why.…

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Q: Let f:A at f is surjective. B and g: B -C be two functions. If gof is surjective and g is total…

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Q: ) Let f: A-B be a function which is not surjective and g : B C which is bijective. Prove that g of:…

A: See full proof below

Q: Let f (x, y) = e-(x²+y²), Find the maximum and minimum values of f on the rectangle given below.…

A: We have to find the maximum and minimum value of the given two variables function.

Q: Suppose the function f has the property that there exists a number B such that |S(x) – f(c)| < B|x –…

A: Suppose the function f has the property that there exists a number B such that |f(x)-f(c)|≤B|x – c|…

Q: Consider the function f : [a, b] → (a, b] with ƒ being continuous. Show that there is a number c in…

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Q: Consider the function θ:{0,1}×N→Zθ:{0,1}×N→Z defined as θ(a,b)=a−2ab+bθ(a,b)=a−2ab+b. Is θθ…

A: Consider the given function as θ:0,1×N→Z defined as θa,b=a-2ab+b. Check for invective as follows: To…

Q: 5. Determine whether f is a function from Z to R if a) f (n) = ±n. b) f (n) = v n2 + 1. c) f (n) =…

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Q: Let f : A → B be a function between sets A and B. Let X1, X2 ⊆ A and Y1, Y2 ⊆ B. a) Prove that if…

A: 1. A collection of well-defined objects or elements is called a set. If a is an element of a set A,…

Q: Let f: {4,5,6} → {1,2,3} be a function defined by f(x) = x-3. Prove that f is injective and…

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Q: Consider the function f : Z × Z → Z × Z defined by f(a, b) = (a+ b, a b) -

A: