Assume f,g : [a, b] → [c, d] where a < b and c < d. Prove that sup{f(x)+g(x) : x € [a, b]} < sup{f(s) : s€ [a, b]} + sup{g(t) : t E [a, b]}. Give complete justification in your proof.

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: f(x) = X-3 15 A) C) for xU for x ≤0 ++D 2 4 6 X 6 X B) D) ←+H NA 67x. -2- 2 4 6 M

A:

Q: Premises: Vx(L(x) → H(x)) 3x (~ J(x) A L(x)) Conclusion: 3x (H(x) A - J(x))

A: Given that Premises:      ∀xLx→Hx                        ∃x~Jx A LxConclusion:  ∃xHx A ~Jx We have…

Q: 18. Determine whether each of these functions is a bijec- tion from R to R. a) f(x) = -3x + 4 b)…

A: As per guidelines, we will solve first three subparts of the question. a) Given function is fx=-3x+4…

Q: (-х+а, х < —1 If f(x) = {x² + 5, –1 < x < 2 is continuous then х — b, х 2 2 a + b =?

A:

Q: Let f(x, y) = (2x - y,x-2y); (x, y) E R x R, (R is set of real numbers.) %3D a) Show that f is one…

A:

Q: 4. Let A and B be sets and let T C B. Also let f : A → B. (a) Use a diagram to show that f(f-'(T)) C…

A: According to the given information,

Q: Let f e E(0, ox) and let a be a positive real number. Define the function fa as follows Sal) = {";…

A: Properties of Laplace transformation

Q: Q₂i- If show that utt ч (хо) ut (x,0) - a 9/x). Z 3 u (X, Z) 2 Ux(xt) = a as 1x1_-2 ux 9. 2 u XX…

A: Using D'Alembert solution for one dimensional wave equation.

Q: x4 – 10 хе (-о, л — е) 2: х4 +9 хе [л— е, оо) - f(x) = Evaluate f(0)г

A:

Q: a) f(AUB) = f(A) U ƒ(B). b) (i) f(An B) ≤ f(A)n ƒ(B). (ii) Find a function f {1,2} → {a}, and A, B C…

A: a)  We  have  to prove  that  Now,      Thus,  (Proved)[since, b) i) We  have  to show  that Let  be…

Q: Find a ≤ b such that

A: ∫abx2-9dxApply the Sum Rule:…

Q: , Consider the interpolating polynomial for f(x) =x² based on th points (0,0). (1,1) and (2,8). Find…

A:

Q: Q2\A\ Which of the following functions are bijective 2- f: R{0}→ R\{0},f(x) = 1/x این 2- f:R→R,f(x)…

A:

Q: Exercise 4. Let F and G be homothetic functions in two variables. Is their sum homothetic? [Hint,…

A: We will use the fact that the MRS of a homothetic function is always homogeneous of degree 0. Let…

Q: 1. Define f : (- o, 2) → (- 0, 1) by f(x) = . Show that f is bijective. x+1 *-2

A:

Q: 5. Let f be defined for all real numbers and suppose f(x)-f(y)| ≤ (x - y)² for all real numbers, x…

A:

Q: Consider the function ƒ (x, y, z) = 2x² + 5y² + z². a) If ƒ (x, y, z) is the profit made from…

A:

Q: Give an example of a function f : (X1, d1) → (X2, d2) that sends compact sets to compact sets, but…

A:

Q: Suppose f: X→Y, and A, B CY. Prove that (a) f-¹(AUB) = ƒ−¹(A) U ƒ−¹(B). (b) f-¹(ANB) = f¯¹(A) n…

A:

Q: Suppose that A,B⊂S such that Pr[A|B]=25, Pr[A]=12 and Pr[B′]=12. What is Pr[B|A′]?

A:

Q: Is the function f(x) = x + 1 from the set of integers to the set of integers onto?

A: For onto from set A to B, Every value of set B should exist atleast once in the function.

Q: 13. Let the function f: [a, b] in R be continuous and one-to-one and such that f(a) &lt; f(b). Let c…

A:

Q: Consider the function f : [−2, 6] → R, x→ f(x) = 2x3/ [√(x2+8)+2]. i) provide argument why this…

A: This is the question of Real topology, and real analysis.

Q: Q.10. Let [a, b] be an interval of R. Let f:[a,b]→R be a function such that f is M-times…

A:

Q: consider the function f(2)= x³ +32+1 in the interval [-3,3]; we can say a) f has no zeroes in the…

A:

Q: (2) Let f(x) = ax³ + bx? + cr + d, where a, b, c, d e R and a + 0. Prove that f(x) must have a root.…

A:

Q: 3. Construct a bijection f: [0, 1] → (0, 1) UN.

A: