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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- Find the value of x for the following sets of congruence using the Chinese remainder theorem.a. x ≡ 2 mod 7, and x ≡ 3 mod 9b. x ≡ 4 mod 5, and x ≡ 10 mod 11c. x ≡ 7 mod 13, and x ≡ 11 mod 12Prove by Contradiction: For all integers x and y, x^2 - 4y is not equal to 2Prove that ⊢ (∃x)(A → B) → (∀x)A → (∃x)B.
- Prove or disprove each of the following Θ(log n)^k = O(n) where k is any constant larger than 1, hint: what is the derivative oflnk n?is ¬(p -> q) logically equivalent with p ^ ¬q? Show your solution.Suppose we have positive integers a, b, and c, such that that a and b are not relatively prime, but c is relatively prime to both a and b . Let n = s × a + t × b be some linear combination of a and b, where s and t are integers. Prove that n cannot be a divisor of c. Follow the definition of relative primes, and use contradiction.
- show that t ↔ ~q is logically equivalent to (t v ~q) → (~t ∧ q) without using a truth table.Suppose that the equation ax b .mod n/ is solvable (that is, d j b, whered D gcd.a; n/) and that x0 is any solution to this equation. Then, this equation has exactly d distinct solutions, modulo n, given by xi D x0 C i.n=d / fori D 0; 1; : : : ; d 1Q1/Simplify the following Boolean functions in products of sums using K-map: a) F(w, x, y, z) = Sigma(0, 2, 5, 6, 7, 8, 10) b) F (A, B, C, D) = Pi(1, 3, 5, 7, 13, 15)