Prove that Note: H(XY) ≤ H(X) using log sum inequality. -X and Y are random variables -X and Y are independent
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- Nelson-Oppen Use the Nelson-Oppen procedure to prove that the following formula over the theories of uninterpreted functions and linear real-valued arithmetic is unsatisfiable: g(f(x – 2)) = x + 2 A g(f(y)) = y – 2A (y +1= x – 1)ab Show that if f(x) then fa - fb = f b-aFor the function f: Z Z prove or disprove whether f is injective and/or surjective f(x) = 4x –6 %3D
- 3 : Prove that the sum of the three variables of the sum exponent f(x) = Σe is strictly convex. For this i=1 purpose, compute the Hessian matrix of the function f(x).Consider nonnegative integer solutions of the equation x1+x2+x3+x4+x5+x6=30. How many different solutions are there? How many solutions also satisfy: for every i∈{1,2,3,4,5,6}, xi is positive and even?By contradiction, prove that ∀ x, y ∈ Z, if XY is even and y is odd then x is even.
- Explain the Wronskian determinant test. Using the Wronskian determinant test, write the program using NumPy to determine whether the functions f(x)=e^(- 3x), g(x)=cos2x and h(x)=sin2x are linearly independent in the range (-∞, + ∞). #UsePythonOptimize the following Boolean function by finding all prime implicant andessential prime implicant f(W,X,Y,Z) =∑ m(0,2,3,5,7,8,10,11,12,13)f(n) = 2", g(n) = 2.01". v.