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 that R is an equiavalence relation

Q: Prove that there does not exist any function f(x, y) such that

A: To show that there does not exist any function that satisfies the given conditions

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…

A:

Q: Let f: A→ X X Y be defined as f (a) = (f₁ (a). f2 (a)) each a ≤ A, Then f is continuous if and only…

A:

Q: Determine if the function f : RxR → R, defined by f(x, y) = x2", for all (x, y) E R×R, is…

A: f is not one-one  But, f is onto.

Q: Let f be a differentiable function. Consider the relation = x•f where z is implicitly defined in…

A:

Q: Use the Product Rule twice to prove that if f, g, and h are differentiable functions of x, then

A: Let us assume f(x)*g(x) together. and h(x) separate.  Then apply the product rule

Q: Let f be a real valued function defined on [a, b] . Let f have a local minimum at a point x ∈ (a, b)…

A:

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: Theorem 11. Let f be a continuous function on R such that f(x+y)3Df(x)+f) x, yER. Show that f(x)3Dcx…

A: Since you have posted a multiple questions. So,  we will solve only first question  for you. To get…

Q: Let f : R → R be a function such that |f (x) – f(y)| 0. Write the e-d definition of continuous…

A:

Q: Prove the following statement: For any function f : R" - R and any function g : R" R, let h: R" R be…

A:

Q: Let f:X → Y be function and g: X → fcX ×Y, g(x) = (x, f(x)). Prove that if g is a homeomorphism then…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Prove that L(f) ≤ 0 ≤ U(f). Given that the bounded function f : [a, b] → R has the property that…

A: Given that the bounded function f : [a, b] → R has the property that f(x) = 0 for all x ∈ [a, b] ∩…

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

A: The relation R is defined by xRy if x and y share a common factor. We need to check reflexivity and…

Q: Let f : R² → R² be a continuous function. Prove that f([0, 1] × [0, 1]) is compact.

A: It is given that f:ℝ2→ℝ2 be a continuous function. Prove that f0, 1×0, 1 is a compact set. Let Gi:…

Q: Show that for any function f : X → Y , there are two functions g : S → T and h : X → S such that (a)…

A: To Show: Show that for any function f : X → Y, there are two functions g: S → T and h: X → S such…

Q: Let X be any set and let Y = "2¬X = {x : X → {0, 1}}, that is, Y is the set of all functions from X…

A: To prove that F is a bijective mapping,we have to show that it is both one-to-one and onto.

Q: b) If the function f(x), g(x) and h(x) are continuous on [a, b] and derivable on (a, b), show that…

A: The given functions f,g and h are continous on [a,b] and differntiable on (a,b). So intermediate…

Q: Let R be the relation on R defined by R= {(x, y) E R × R | x² + y² = 4 or x = y²}. Answer each of…

A: Given that R be the relation on ℝ defined by,R=x,y∈ℝ×ℝx2+y2=4 or x=y2

Q: Let R be a relation from A={−1,0,1,2} to B={a,n,i} such that (x,y) provided that x is non-negative…

A:

Q: Let g : R → R be a positive function such that g(0) = 1 and g(x+y) = g(x)g(y), ∀x, y ∈ R. Show that…

A:

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: Let f:X → Y and g:Y → Z are functions show that a) If g of is surjective then g is surjective b) If…

A:

Q: Let A = {x, y} and B = {w,z}. Describe all relations from A to B. Indicate which of these are…

A:

Q: 3. If f:(X,r)(Y.A) and g:(Y,A)→(Z,r) are open functions, prove that gof :(x.r)(Z.r) is open…

A: Definition: An open map is a function between two topological spaces that maps open sets to open…

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: Let f:Z x Z+ Z x Z be the function given by f(z, y)= (2z + y,I- y). Find, with proof, an element af…

A:

Q: Define a relation C from R to R as follows: For any (x, y) ∈ R x R, (x, y) ∈ C means that x² + y² =…

A: (x,y) ∈ R × R (x,y) ∈ C means    x2 + y2 =4

Q: Let f : R → R be continuous such that f(q) = 0 for all q ∈ Q. Prove that f must be the zero function

A:

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

A:

Q: We consider a function f from R to R such that f(0)=5. Show that f cannot be a linear map. Hint:…

A:

Q: Let p : X → Y and f : Y → X be continuous. Suppose that po f equals the identity map of Y. Prove…

A:

Q: Let X be any set and let Y = 27X - {x : X {0, 1}}, that is, Y is the set of all functions from X to…

A:

Q: Let f: X→Y be a function. Please prove that if f(X-A)=Y-f(A) for any A⊆X, then f is onto.

A:

Q: et f: (a, b) → R. If f'(x) exists for each x, a < x <b, prove that f is continuous there.

A:

Q: Let R be a relation from X to Y and S be its reverse relation from Y to X. Prove or disprove that R…

A: According to the given information it is required to prove/disprove that R is injective and well…

Q: Let f be a function that is defined for all x and that satisfies the equation f (x + y) = f (x) · f…

A:

Q: Let f be defined on [a, b], if f has a local maximum at a point x E (a, b), and if f'(x) exists,…

A:

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

A:

Q: Let f:R" → R be a continuous function, and let x1, x2 € R" be distinct. Prove that there exist a…

A: Given: Now consider the given, Let f:ℝn→ℝ be a continuous function, and x1, x2 ∈ℝn  be two…

Q: Let f : R → R be such that for all x, y ∈ R, |f(x) - f(y)| ≤ (x - y)2. Prove that f is a constant…

A:

Q: Suppose f : [a,b] → R is a continuous function. Prove that the direct image f([a,b]) is a closed and…

A:

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

A:

Q: A function F from A to B is a relation from A to B that satisfies two properties: 1. for every x E…

A:

Q: Let fi and f2 be concave functions from R" to R. Let f : R" +R be defined as the pointwise minimum…

A:

Q: B. Show that the space V = {f : R → R} of functions from R to R is not finite- dimensional.

A: Let V=f:ℝ→ℝ. Let A is set of all polynomial with integer coefficient. We know that every polynomial…

Q: Determine whether the following relation is a function from R into R. A Iso deter- mine whether that…

A:

Q: Suppose f : X → Y is a function. Prove that f is surjective if and only if for all sets Z and all…

A: