1. Consider the function f : Z × Z → Z × Z defined by f(a, b) = (a+ b, a – b). Is f injective? Is f surjective? Justify your answers.

Q: 3. Let f(x) = x² + 3 and g(x) = Vx – 3 be functions defined on [0, ∞) and [3, 00), respectively. i.…

A: According to question,  Given functions, f(x) = x2+3, g(x) = x-3 Function fx defined on interval :…

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

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Q: Q: Show that the functi on f:R →R given by f (x) = ax +b, where ab e R, a #0 is a bijection.

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Q: c) Let the set A = {-1,0,1} and consider the following function: ƒ3 : A × A → A × A given by f(x, y)…

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Q: Consider f : R → R given by f(x) = x². What are the domain, codomain and range of f?

A: Let a function f : A→B thenDomain: set ACodomain: BRange: Values that comes out after putting values…

Q: Let f: XY and A,BCX. Give an example of a function f and sets A and B that f[A\B] #f[A]\f[B]

A: Take any constant function and subsets A and B such that they overlap.

Q: Suppose f X→ Y, and A, B C X. Prove that (a) f(AUB) = ƒ(A) U ƒ(B) (b) ƒ(ANB) ≤ ƒ(A) Ñ ƒ(B)

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Q: Definition: Let f : D → Y be a bijection. The inverse function f-1 : Y → D is a function such that…

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Q: Consider a function f whose domain is the interval [a, b]. Show that if |f(x) − f(y)| ≤ (x − y)²,…

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Q: Let R = {(n, −n) | n ∈ Z and n ≤ 0} ∪ {(n, n + 1) | n ∈ N}. Is R a function? Justify your answer.

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Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

Q: 3. Let f: R³ R be defined by f(x, y, z) =|| x ||. Find Vf(x, y, z). What is the domain of Vf?

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Q: Let f : R → R be a function defined by f(x) = Decide whether it is an injection, a surjection, 1+ x2…

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Q: Evaluate fff f(x, y, z) dV for the specified and B: B f(x, y, z): Z | 8 2 ≤ x ≤ 6,0 ≤ y ≤ 6,

A: We have to find the value of the given expression according to the given information

Q: True or False: ⊨ F ⊃ T ⊨ - means logically valid , F-false, T-true

A: We know,       A ⊃ B is equivalent to ~A v B

Q: Consider the function h : Z xZ → Z× Z defined by h(x, у) 3D (2ху, 7x — Зу). (a) Is h injective? (b)…

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Q: f :R →R, f(x) = 3 + ax² + bx +11 fonksiyonu (-3,5) aralığında 20 - azalan ise atb-kaçtır ? If the…

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Q: Let the function f: R R be defined by f(r) = 3r -5. Prove that the function is bijective.

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Q: Suppose f: X→ Y, and A, B CY. Prove that (a) f¹(AUB) = f¹(A) U f¹(B).

A: Union of two sets A and B is denoted as A∪B  It contains all the elements that are either in A or in…

Q: Determine the domain and range of the function f: {r,s,t,u} ➔ {A,B,C,D,E} defined by f(r) = A, f(s)…

A: Domain of a function is the set of possible values of independent variable. The domain of given…

Q: Let f : A → B be a function. Prove that if C C A and D C A and f is injective, then f(C) = 7 (D) =→…

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Q: Show that the function f: N² → Z given by f(m, n) = m²-n² is neither injective: suriective. nor

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Q: A function f: Z× Z → Z is defined as f(m,n) = 2n-4m. Verify whether this function is injective and…

A: Here we have given a function f , we have to check, is this function injective and surjective?

Q: . Consider the function f: Z x Z → Z × Z defined as f(n, m) = (n +m, 3n + 4m). (d) Find the inverse…

A: I have find two equation in variables m & n and solved them