Consider the relation on functions f: R → R given by f g if there are numbers A, B E R such that for all x, f(x) ≤ Ag(x)+B. Is this an equivalence relation?

Q: Determine if the relation ? = {(f, g)|f(x) = g(x) for some x ∈ ℝ} on the set of all functions from ℝ…

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Q: 4. Let f : X → Y define a relation on the set X by „Ry if and only if (x) = fy). Is f an equivalence…

A: f is an equivalence relation if f satisfies three properties. 1. if f is a reflexive relation  2. if…

Q: Consider the following relation < over reals: x < y iff (x– y) e Z. Prove that it is an equivalence…

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Q: Consider the relation R on the set of functions defined by (f, g) = R if and only if f(x) is…

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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: 1. Let R be a relation on R x R such that (*1vi)Rs02) if and only if x,yı = x2y2. (a) Is (2,3)R(1,4)…

A: A relation from a set A to B is defined as a subset of A×B. Here R is a relation on ℝ×ℝ such that…

Q: 5. Let f: A B be a function. Define a relation on A by a₁ ~ƒ a₂ ⇒ f(a₁) = f(a₂) for all a₁, a2 € A.…

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Q: Consider the relation R = {(a, b) ∈ Z × Z : 3a − 5b is even }. Show that R is an equivalence…

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Q: Let RC RX R with {(x, y) |x² =5 y²}. Characterize R in terms of whether it is reflexive,…

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Q: Let A and B be two nonempty sets, f: A → B be a function and R be an equivalence relation on the set…

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Q: R.2. Consider the relation R=- ={x,y):x.yeR&]x=}<3}. Is R an equivalence relation on set of real…

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Q: H 22. D is the relation defined on Z as follows: For all m, n = Z, m Dn ⇒ 3/(m²-n²).

A: A relation S is said to be an equivalence relation on set A if it is reflexive, symmetric and…

Q: Let f : A → B be an injective function, a be an order relation on B and 3-1 = {(x, y) E A x A :…

A: An order relation means the criterion of comparison between objects, that satisfies the properties…

Q: Let f : R → R be defined as f(x) = x3 + 8 (i) Prove that f is a bijection (one-to-one…

A: Consider the given function.

Q: f: X → Y be a surjective function. Define a relation on X by setting ay iff f(x) = f(y). Show that…

A: Please check the solution in next step

Q: f(y). (a) Prove that is an (b) Find a bijection from A/~ to B. equivalence relation.

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Q: (29) Let~ be the relation on R given by y Prove that is an equivalence relation on R. Find [].…

A: A relation ~ on the set of real numbers R is defined as x~y if and only if x-y∈Z. Our aim is to show…

Q: Question 5: Check whether the relation R in R defined as R = {[a, b): as b} is reflexive, symmetric…

A: Introduction :--

Q: 4. Describe all Consider the relation R on R x R defined by (x1, y1) R(x2, y2) iff x₁ - 2y2 = x₂ -…

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Q: Let f : A B be function. Prove that the relation a - b if and only if f(a) = f(b) is an equivalence…

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Q: Let F= {f|f: R → R}, and define a relation R on Fas follows: R = {(f,g) e F xF | 3h e F(f = h o g)}.…

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Q: Define S to be the following relation on R[x]: S = {(f, g) = R [x]²: (x + 1) divides (g - f) in R[x]…

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Q: Check if the following statement is TRUE or FALSE. Let R be the relation on R² defined by…

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Q: Let f : X → Y be a function. Define a relation R in X given byR = {(a, b): f(a) = f(b)}. Examine…

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Q: A relation R on R² is defined by ((x1, y1), (2, 2)) R if Prove that R is an equivalence relation.…

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Q: Prove that these relations on the set of all functions from Z to Z are equiva- lence relations.…

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Q: F U}. Deine an equivalence relation on S by: (a, b) ~ (c, d) if and only if ad – bc = 0. (a) Show…

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Q: Let R be the relation on N defined by x R y if x and y share a common factor other than 1. Determine…

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Q: 4. Let the relation R = { (a, b) e Z × Z l(a + b) = 1 (mod 5) }. a. Is the relation transitive?…

A: A relation R between two sets A and B is the set of all ordered pairs a,b such that a∈A and b∈B. We…

Q: Let S be the following relation on R: S = {(x, y) = R² :y-x is rational}. Prove that S is an…

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Q: 2. There is a function f : X →Y . Let's define a relation R in a set X, by a formula: (xRy) → (f(x)…

A: Given f: X → Y and R is the relation in the set X such that x R y ⇔ f(x) = f(y)

Q: R.1. Consider the relation R = {(x,y): x,y eR & x² = y*}. Prove that R is an equivalence relation on…

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Q: Let f : A → B be a function and let G be an equivalence relation in B. Prove that f (G) is an…

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Q: 4|(x+3y)

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Q: Define an equivalence relation on R² by: ¤1, Y1) ~ (x2, Y2) e if and only if (*1, Yı) and (x2, Y2)…

A: The complete solutions are given below