Let C and D be finite sets and let f: C D and g: D C be two functions such that Vy eD f(g(y)) = y Prove that C| > \D].

Q: f,(x)= e* f2(x)=e¯* (x) = e'

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Q: Let h: R → R be a continuous function on all of R. If h(x) Tt, prove that h(t) 50.

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Q: Show that the subset of R2 given by {(x, y) E R² : xy 1, x > 0} is closed.

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Q: Prove that the function f : R → R, given by f(x) = 2x, is one-one and onto.

A: Given, function f : R → R, given by f(x) = 2x.

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…

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Q: Let X, Y be two sets. Prove that X × Y C P(P(X UY )).

A: It is given that X,Y  be two sets. To show that X×Y⊆PPX∪Y Firstly, we define the set X×Y as, X×Y=x,y…

Q: Consider a bounded function f [a, b] → R and a partition P = U(f, P) ≥ U(f, Q) and L(f, P) ≤ L(f,…

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Q: 2. Let a, b e R and let f be the function Z + Z defined by f(z) - ar + b. For which values of a and…

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Q: Show that the subset of R2 given by {(x, y) E R² : xy = 1, x > 0} is closed.

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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Q: If  and f is continuous, then  f must have at least one zero between  a and b. (Assume a ≠ b.)

A: Case1: Whenfx=0 , then ∫abfxdx=0 is obvious.

Q: (b) Show that the u(x, t) can be written as u(x, t) = 0(x – at) + 4(x + at), where o and are…

A: The multivariable function u(x,t) is a rule that associates or fixes a number to a particular point…

Q: 2. Let f be defined on an open interval (a, b) and assume that for some e e (a, b) we have f'(c) > 0…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/xe A}. 1…

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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: Let f and g represent differentiable functions such that f ″ ≠ 0 and g″ ≠ 0.

A: Concavity of a function can be determined using the second derivative of the function. If the second…

Q: t, 0<t<1 f(t) -1, 1, 1<t< 2 || –t, 2<t<10

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Q: If X and Y are finite sets, prove that XY = |X|Y. Here, XY is the set of all functions from Y to X.

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Q: Let B be the set of all bounded sequences of real numbers and define the function d: B xB +R by d(x,…

A: In the above question it is given that B is the set of all bounded sequences of real numbers and a…

Q: Let f be the function from {a, b, c, d} to {1, 2, 3} defined by f(a) = 3, f(b) = 2, f(c) = 1, and…

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x € R: 1/x e A}. 1…

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Q: Prove that f (x) = x-12 is : (i) not sectionally continuous in every finite interval in the

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Q: Find all values of m such that a(r) = {X – 1 if x m is continuous everywhere.

A: Questions from real analysis.

Q: Suppose that f:[a,b] R is bounded, P is parition of [a,b] and Q is refinement ofP, then: L(f;P) <…

A: Answer

Q: A function from ZxZ to Z defined by f(m, n) = m-n is an onto function. True False

A: Here we will examine that f is onto or not.

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: Let X, Y be sets and f : X →Y. Prove f"(A\B) = f"(A) \ f¯"(B). for any A, B CY.

A: Solution:Given:          X, Y    are the Sets and                      f: X→ Y To prove:  f-1(A \…

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

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Q: Let (X, A, µ) and (Y, B, v) be measure spaces. Prove that (a) if E E A x B then y-section of E is…

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Q: Define f: R → [2, 0) by f(x) = x² +2,Vx € R.Prove that f is onto.

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Q: Let A be a non-empty set of positive numbers, which is unded above and let B = {x e R: 1/xe A}. 1…

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Q: 7. Show that the function f : R → R* defined by f(x) is not onto. x2 +1

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Q: Let A be a non-empty set of positive numbers, which is bounded above and let B = {x e R: 1/x e A}. 1…

A: Remark: Completeness property of R: Every bounded above non-empty subset of R has a least upper…

Q: The function on ℤ defined by f(x)=-x^2-5 is onto. True or False ?

A: The given function is                  f(x) =x2-5   Graph:

Q: Let f: R → R with the usual topology that sends compact sets to compact sets in R ^ 2 Is the…

A: For any x,y∈R if y=fx there there exists a number x±∈R in neighborhood of x such that y±=fx±.

Q: Let X, Y be two sets. Prove that X x YC P(P(X UY )).

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Q: Let f(x) x² and g(x) = xf(x). Find c = (0, 1) as guaranteed by the Cauchy Mean Value Theorem.

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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: The relation xy >= 1 where x,y are on the set of all integers is reflexive. True False

A: Given relation xy≥1, where x,y are on the set of all integers

Q: If f(x) is continuous on a closed interval I, and f(a) and f (b) have opposite signs where a and b…

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Q: |A:k||2||Br:||2) < ||A||F||B||F k=1

A: answer is in next step

Q: Let S«) = {r²+2x}°, 121, ax+bx, x< 1. Compute for all a and b such that the function f(x) is…

A: We have to find

Q: Determine whether or not the function f : Z × (Z − {0}) → Z is onto, if f((m, n)) = ⌊m/n ⌋

A: Given that: f(m,n)=mn,  f:Zx(Z-{0})→Z

Q: Give an example of a continuous function T, and an open set O in a metric space (X, d) such that…

A: Consider the provided question, Hello. Since your question has multiple parts, we will solve first…

Q: Q1- If f is bounded and define on [a, b] then show that m(b – a) < L(P,f)< U(P,f) < M(b – a). Q2-…

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Q: Let f(x) = x² and g(x) = xf(x). Find c = (0,1) as guaranteed by the Cauchy Mean Value Theorem.

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Q: Let (X, Tx), (Y, Ty) represents topological spaces such that X = {a,b,c,d}, Tx = {X. Ø, {a}, {a, b},…

A: CONTINUOUS FUNCTION: If the inverse image of every open subset of Y is open in X, then the function…

Q: Let A be a non-empty set of positive nunıbers, which is bounded above and let B = {x e R: 1/x e A}.…

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