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using discrete structures 3. Consider the function F(x, y, z) for x, y, z z 0 defined as follows: a. F(x, y, 0)-y+1 b. F(x, 0, 1)-x c, F(x, 0, 2) = 0 d. F(x, 0, z+ 3)-1 e. F(x, y, z)-F(x, F(x, y-1, z...
(1 point) A function f is defined on the whole of the x, y-plane as follows: f(x,y)0 fy0 otherwise For each of the foll...
(1 point) A function f is defined on the whole of the x, y-plane as follows: f(x,y)0 fy0 otherwise For each of the following functions g determine if the corresponding functionf is continuous on the whole plane. Use "T" for true,"F" for false 2. g(x, y) 9x2y 3. gx, y)-4 sin) 4. g(x, y) xy sin(xy) 5. g(x, y) 3xy
(1 point) A function f is defined on the whole of the x, y-plane as follows: f(x,y)0 fy0 otherwise For...
2. (a) Prove by structural induction that for all x E {0,1}*, \x = x. (b)...
2. (a) Prove by structural induction that for all x E {0,1}*, \x = x. (b) Consider the function reverse : {0,1}* + {0,1}* which reverses a binary string, e.g, reverse(01001) = 10010. Give an inductive definition for reverse. (Assume that we defined {0,1}* and concatenation of binary strings as we did in lecture.) (c) Using your inductive definition, prove that for all x, y E {0,1}*, reverse(xy) = reverse(y)reverse(x). (You may assume that concatenation is associative, i.e., for all...
Consider the bivariate function f(x.y) = (x + y)/3 for 0< x< 1 and 0<y< 2...
Consider the bivariate function f(x.y) = (x + y)/3 for 0< x< 1 and 0<y< 2 and f(x.y) = 0 3. otherwise. (a) Show that f(xy) is a density function. (b) Find the probability that both X and Y are less than one. (c) Find the marginal densities of X and Y and show that they are not independent. (d) Find the conditional density of X given Y when Y = 0.5.
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0...
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
Discrete Structures A={ab,bt,cq,qr,rc,ta,zz} 1. Consider the following permutation function P, (a b c d e f...
Discrete Structures
A={ab,bt,cq,qr,rc,ta,zz}
1. Consider the following permutation function P, (a b c d e f g h i ghi e fac db) a. If P. is a permutation function on the set A, can you determine the set A with certainty (if so, write it below)? b. Represent P, as a set of ordered pairs. C. Find the digraph of P, d. Is P, a relation? e. Can a permutation function ever not be a relation? f. Is P,...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x,...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
Discrete Structures Name: Problem 2. Prove the following theorem using P Theorem. Let x, y e...
Discrete Structures
Name: Problem 2. Prove the following theorem using P Theorem. Let x, y e Z. If c-y is odd, then 1 em using proof by contrapositive. yis odd, then ris odd or y is odd.
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 :...
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y)...
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
real analysis II. Consider the function f:[0,1] - R defined by f(x) 0 if x E...
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
(1 point) A function f is defined on the whole of the x, y-plane as follows: f(x,y)0 fy0 otherwise For each of the following functions g determine if the corresponding functionf is continuous on the whole plane. Use "T" for true,"F" for false 2. g(x, y) 9x2y 3. gx, y)-4 sin) 4. g(x, y) xy sin(xy) 5. g(x, y) 3xy
(1 point) A function f is defined on the whole of the x, y-plane as follows: f(x,y)0 fy0 otherwise For...
2. (a) Prove by structural induction that for all x E {0,1}*, \x = x. (b) Consider the function reverse : {0,1}* + {0,1}* which reverses a binary string, e.g, reverse(01001) = 10010. Give an inductive definition for reverse. (Assume that we defined {0,1}* and concatenation of binary strings as we did in lecture.) (c) Using your inductive definition, prove that for all x, y E {0,1}*, reverse(xy) = reverse(y)reverse(x). (You may assume that concatenation is associative, i.e., for all...
Consider the bivariate function f(x.y) = (x + y)/3 for 0< x< 1 and 0<y< 2 and f(x.y) = 0 3. otherwise. (a) Show that f(xy) is a density function. (b) Find the probability that both X and Y are less than one. (c) Find the marginal densities of X and Y and show that they are not independent. (d) Find the conditional density of X given Y when Y = 0.5.
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if r 0 0 if r <0 θ(z) =
2. Prove the following useful properties of Dirac δ-functions (a) δ(ax) = (z) (b) zfic) =0 (c) f(x)5(-a) f(a)5(r d) δ(z-a (aメ0) a) ( dz9(x-a) ) where θ(x) is the step function defined as 1 if...
Discrete Structures
A={ab,bt,cq,qr,rc,ta,zz}
1. Consider the following permutation function P, (a b c d e f g h i ghi e fac db) a. If P. is a permutation function on the set A, can you determine the set A with certainty (if so, write it below)? b. Represent P, as a set of ordered pairs. C. Find the digraph of P, d. Is P, a relation? e. Can a permutation function ever not be a relation? f. Is P,...
5. [12 Marks) Consider the level surface of the function f(x, y, z) defined by f(x, y, z) = x2 + y2 + x2 = 2a?, (1) where a is a fixed real positive constant, and the point u = (0,a,a) on the surface f(x, y, z) = 2a. a) Find the gradient of f(x, y, z) at the point u. b) Calculate the normal derivative of f(x, y, 2) at u. c) Find the equation of the tangent plane...
Discrete Structures
Name: Problem 2. Prove the following theorem using P Theorem. Let x, y e Z. If c-y is odd, then 1 em using proof by contrapositive. yis odd, then ris odd or y is odd.
(2) Let f(z, y)-xy +x-y be defined on the closed disk {(z, y) E R2 : z? + y2 < 4} of radius 2. (a) Find the maximu and minimu of Duf at (0,0) over all unit vectors u. (b) Find the maximum and minimum of Duf over all points in the disk(,y) E R2 r2 + y2 < 4} and all unit vectors u. (llint. Think of IvJF as a function ofェand y in the disk.)
2. Consider the function f : R2 → R defined below. r3уг_ if (x,y) (0,0) f(x,y) = if (x, y) (0, 0) (a) Prove that f is continuous at (0,0) (b) Calculate the partial derivatives (0,0) and (0,0) directly from the definition of partial derivatives. (c) Prove that f is not differentiable at (0,0).
real analysis
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest terms. 1. Prove that f is discontinuous at every x E Qn [0,1]. 2. Prove that f is continuous at every x e [0,1] \ Q.
II. Consider the function f:[0,1] - R defined by f(x) 0 if x E [0,1]\ Q and f(x) = 1/q if x = p/q in lowest...
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