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

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: . 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. Show that which of these relations on the set of all functions on Z→Z are equivalence relations?…

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A: Please check the solution in next step

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Q: : Let f [a, b] → R be a continuous and one-to-one function, where a < b are real numbers. Use the…

A: Introduction: There must be at least one "c" such that c∈(a,b) (or a&lt;c&lt;b) and f(c) = L. if…

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Q: Given that f is a function from A onto B, that g :B →C, and that the function g • f is one-one,…

A: In the given question we have to find f and g are one - one function.

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Q: Assume (r1, y1) ~ (x2, Y2) → f(x1,Y1) = f(x2, Y2). Prove that ~ an equivalence relation.

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Q: Every function is a relation of its domain and codomain. O True. O False.

A: A relation defined between two sets X and Y contains ordered pair a,b such that  a∈X ,  b∈Y. It…

Q: Let Q: A - B be a function and CSA. define the restriction and also the extension of a function

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Q: Let X-(3, 4, 6, 8). Determine whether or not the relation R = {(x, f(x)) :x - X, f(x) %3D = x2 + 1}…

A: Solution is given as: Hope the answer is helpful.

Q: Suppose that f : [a,b] → R is a continuous and one-to-one function. Prove that f is strictly…

A: We will prove this by contradiction.

Q: Let f: X →Y be an injective function, and A, B C X. Prove that Q1) f(A)n f(B) C f(AN B).

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Q: Suppose that f : A → B and g : B → C are both injective functions, prove that the function go f : A…

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Q: 1. Show that which of these relations on the set of all functions on Z-→Z are equivalence relations?…

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Q: T is a function?

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Q: Let R be a relation on the set of all functions from Z to Z defined by: R = {(f,9)|f(1) = g(3) or…

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Q: Prove that for functions f : R→R, the e- 6 definition of continuity implies the open set definition.

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Q: Let R be a relation on Z given by aRb if and only if a = b². Is this relation a function? Justify…

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Q: Let f C D be a function, with C being the domain and D being the codomain. Show that for any C₁, C₂…

A: Functions

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Q: Define the relation A=((x,x+ 1):xER] Show that A, is a function from R to R.

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Q: Let f : A → B and g : B C be surjective functions. Prove or disprove whether gof is surjective.

A: Here given that                       f :A→B   and    g :B→C  be both surjective function .  Now we…

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Q: 6. Prove that there does not exist a multiplicative function f such that f(30) = 0, f(105) = 1, and…

A: We have to prove the given statement.

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A: Given:

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Q: Let f : A → B be a function. Prove that f is surjective if and only if, for every pair of functions…

A: Please comment if you need any clarification. IF you find my answer useful put thumbs up. Thank you.

Q: Select ALL that are true about the relation shown below.

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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: 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: 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

Q: Let A be a nonempty set and let f : A→ A be a function. Prove that if f o f = i4, then f is…

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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:…

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Q: Let f : A B be a function. Prove that f is surjective if and only if, for every pair of functions g…

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Q: On P2, define a relation p(x) ~ q(x) if p(0) = q(0). (a) Show that this is an equivalence relation…

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Equations and Inequations

Equations and inequalities describe the relationship between two mathematical expressions.

Linear Functions

A linear function can just be a constant, or it can be the constant multiplied with the variable like x or y. If the variables are of the form, x2, x1/2 or y2 it is not linear. The exponent over the variables should always be 1.