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Consider KB: Vcg F(2, 3). Prove using resolution-refutation that Vxy F(y,x).
5. Prove the following Predicate Logic theorem using resolution refutation • Premise 1. For all persons, a person’s mother is that person’s parent 2. For all persons, if the person’s parent is alive then the parent is older than the person 3. Mary is the mother of John 4. Mary is alive • Conclusion? 1. Mary is older than John Hint: use the following predicates Mother(x,y): x is a mother of y Parent(x,y): x is a parent of y Older(x,y):...
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), z-1) Using Induction, prove the following a. F(x, y, 1)-x +y b, F(x, y, 2) = xy c. F(x, y, 3)-xy
3. Consider the function F(x, y, z) for x, y, z z 0 defined...
prove thsh f(x,y) then n exists 3, 、 2 2.
prove thsh f(x,y) then n exists 3, 、 2 2.
3. (7 points) Consider the function sin f (x, y) = { if (x, y) + (0,0) if (x, y) = (0,0) (a) Prove that f is differentiable at (0,0). (b) Prove that f is not C1 at (0,0). (Hint for part (a): Begin by showing that fx(0,0) and fy(0,0) exist and find their val- ues, and thereby determine Jf(0,0).)
Consider the function f(x, y) = x^3 − 2xy + y^2 + 5. (a) Find the equation for the tangent plane to the graph of z = f(x, y) at the point (2, 3, f(2, 3)). (b) Calculate an estimate for the value f(2.1, 2.9) using the standard linear approximation of f at (2, 3). (c) Find the normal line to the zero level surface of F(x, y, z) = f(x, y) − z at the point (2, 3, f(2,...
2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x)
2. Prove that if X, Y have a joint density, then for any Be B, f(y, x) JB f(x)
Given f(x,y) = 2 ; 0 <X<y< 1 a. Prove that f(x,y) is a joint pdf b. Find the correlation coefficient of X and 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).
Problem 2 Suppose two continuous random variables (X, Y) ~ f(x,y). (1) Prove E(X +Y) = E(X)+ E(Y). (2) Prove Var(X + Y) = Var(X) + Var(Y)2Cov(X, Y). (3) Prove Cov(X, Y) E(XY)- E(X)E(Y). (4) Prove that if X and Y are independent, i.e., f(x, y) Cov(X, Y) 0. Is the reverse true? (5) Prove Cov (aX b,cY + d) = acCov(X, Y). (6) Prove Cov(X, X) = Var(X) fx (x)fy(y) for any (x,y), then =
[5 marks, 2, 3 marks respectivelyl Use the deduction theorem and resolution (but NOT Post's theorem) to prove the following: 3.
[5 marks, 2, 3 marks respectivelyl Use the deduction theorem and resolution (but NOT Post's theorem) to prove the following: 3.