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

Q: Let f(x) be defined for a ≤ x ≤ b. Assuming appropriate properties of continuity and derivability,…

A:

Q: Let In, f be real fun ction on 52.Show {w : fu(w) yo frw) }=Ŭ o U {w-1fu(w) %3D f(w)1>と3

A:

Q: Prove by any method that the interval (-1, 2] is not a compact subset of R. Explain each step in…

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Q: Prove the following Quotient Rule, where f, g are differentiable:

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Q: Let f : R R with f(x) = -2x +5 Prove that f is onto.

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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: Use the Squeeze Theorem to show that f(x) = x² is in R[0, 3]. %3D

A:

Q: Show that the function f : R → R,defined as f(x) = x2 , is neither one-one nor onto.

A: We will show this by counter examples.Let x= 2 and -2 Then f(2)= 2^2 = 4 and f(-2)= (-2)^2 = 4 This…

Q: Find example where of is continuous and an - bijective but f¹ is not

A:

Q: What, if any, are the removable discontinuities of f(x) = ? How do you remove them?

A: Give that: The function f(x) = 1x .

Q: Determine whether the function f: R- R, f(x) =,x # 3 is surjective. x+1 %3D r-3

A: Consider the provided question, We need to determine whether the function f:R→R fx=x+1x-3, x≠3 is…

Q: ) Find a function f(x) satisfying these conditi

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Q: let f: R-->R be given by f(x)= 3x^2 a. Show that f is NOT one-to-one b. is f onto? Explain

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Q: Consider a relation R on Z defined as follows: xRy → xy is even. a. Give a counterexample to show…

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Q: Prove that the function f: R –→ R defined by f (x) = 3x – 5 is bijective.

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Q: Is the relation R, defined on the set of real numbers by xRy if 2x + y = 3. antisymmetric?

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Q: Let f, g,h: A→ A be any bijective functions. Then (f ogoh)-

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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 f(x) = Vx. Compute the difference quotient for f at a = 7 with h = 4. %3D %3D

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Q: Consider the relation S on N defined by uSy + 5|(y + u). Determine whether S is symmetric, reflexive…

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Q: Let A and B be set .Show that AxB → BxA Such that (a,b)=(b,a) is bijective function

A: A function f from a non-empty set A to anther non-empty set B is said to be an one-to-one or…

Q: Define a relation M from R to R as follows: For all real numbers x and y, (x, y) E M means that y =…

A: Use the vertical line test; if you can draw a vertical line anywhere that crosses more than one…

Q: T is a function?

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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 be the function from {a, b, c, d} to {1, 2, 3, 4} with f(a) = 4, f(b) = 2, f(c) = 1, and f(d)…

A: For bijection, the function needs to be one to one and onto.

Q: Show that the function f : R→ R : f(x) = x° is one-one and onto.

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Q: Let f:R-R defined by f(x) =x² , then f ( x) is bijective صواب ihi

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: Define the relation ~ on the set R of real numbers by x~y if and only if x = -y. Show that ~ is not…

A:

Q: Prove ¬(r ∨ ¬(r ∧ s)) is logically equivalent to F using equivalence laws

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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: Show that the function f: Real #s to Real #s defined by f(x)=x2 is not a homomorphism.

A:

Q: Let f(x)= Find f'o. Hint: do not evaluate the in tergral,

A: We will use the Leibnitz formula for differentiation of an integral.

Q: 2|A:k||2|| Bx:||2) < || |||B||F k%3D1

A: Please check the answer in next step

Q: Let f:R-R defined by f(x) =x² , then f(x) is bijective صواب ihi

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Q: Prove that f : R → R by f(x) = 2x+4. Prove that f is one-to-one and maps surjectively onto R.

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Q: Suppose that f : A → B is a function, C C A and DC B. (a) If f is surjective, prove that D = =…

A: Given- Suppose that f : A → B is a function, C ⊆ A and D ⊆ B.  To Prove- If f is surjective, prove…

Q: Show that f : ℝ → ℝ, f(x) = 5x − 3 is a bijective function.

A: The given function is f:ℝ→ℝ defined by fx=5x-3. We say that a function f is bijective if it is both…

Q: Examine the continuity of if 0 a at

A: We need to examine the continuity of the given function f(x) at x=a.

Q: Let f : A B with C, C1, C2 C A and DC B. (a) Show that if f is INJECTIVE (ONE-TO-ONE) then ff(C)] =…

A: "Since you have posted multiple questions, we will answer the first question.  If you want the…

Q: Show that the relation R in R defined as R = {(a, b) : a ≤ b}, is reflexive andtransitive but not…

A: Given, the relation R in R defined as R = {(a, b) : a ≤ b}.

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…

A:

Q: f is a bijective function

A: False: Every function has an inverse We have to find out which statement is false.

Q: Determine whether the function f: Z x Z -→ Z is onto if f (m, n) = .m + n

A: We will solve the problem using the definition of onto function.

Q: Suppose that f is analytic on a domain D andVz € D,f'(z) = af(z) for some constant a. Show that f(2)…

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Q: Take the following derivation rule in predicate calculus: Vr.P(x)Q(x), Vr.Q(x) → R(x) Vx.P(x) → R(x)…

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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: Use FTC I to prove that if |f '(x)| ≤ K for x ∈ [a, b], then |f (x) − f (a)| ≤ K|x − a| for x ∈ [a,…

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Q: i Determine whether or not the relation R on the set of all real numbers is reflexiv netric,…

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Continuous Probability Distributions

Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.

Normal Distribution

Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!