5. Suppose P(m,n) means “m>n”, where the universe of discourse for m and n is the...
Predicates P and Q are defined below. The domain of discourse is the set of all positive integers. P(x): x is prime Q(x): x is a perfect square (i.e., x = y2, for some integer y) Find whether each logical expression is a proposition. If the expression is a proposition, then determine its truth value. 1) ∃x Q(x) 2) ∀x Q(x) ∧ ¬P(x) 3) ∀x Q(x) ∨ P(3)
question 5 5. (a) Informally find a positive integer k for which the following is true: 3n + 1 < n2 for all integers n > k-4 (b) Use induction to prove that 3n +1 < n2 for all integers n 2 k. 6. Consider the following interval sets in R: B-4.7, E = (1,5), G = (5,9), M-[3,6]. (a) Find (E × B) U (M × G) and sketch this set in the-y plane. (b) Find (EUM) x (BUG)...
6. Fix b (a) If m, n, p, q are integers, n > 0, q > 0, and r = mln-plg, prove that Hence it makes sense to define y (b")1/n. (b) Prove that b… = b,b" if r and s are rational. (c) If x is real, define B(x) to be the set of all numbers b', where t is rational and tSx. Prove that b-sup B(r) ris rational. Hence it b" = sup B(x) for every realx (d)...
Roots (20 points). Consider the loop-gain transfer function L(S) = TS-a)n-m where n and m are integers such that n > m and a € R. Also, consider the characteristic equation 1+ KL(S) = 0, with 0 <KER, which can be equivalently written as nam (s– an-m + K = TI (s – rj) = 0. Show that num ri=(n - m), for any 0 <KER.
2. Suppose Xi ~ N(8,02) where θ > 0. (a) Show that s--(x, Σ¡! xi) is a sufficient statistic of θ where X is the sample mean. (b) Is S minimal sufficient? (c) Can you find a non-constant function g(.) such that g(S) is an ancillary statistic?
5. A non-negative valued continuous random variable X satisfies P(X > x +y|X > x) = P(X > y) > 0 for any x,y > 0. (a) Show that P(X > nx) = [P(X > x)]" and P(X > x/m) = [P(X > x)]1/m for positive integers n, m. (b) Show that X~ exponential() for some A > 0.
Let A be a set with m elements and B a set of n elements, where m, n are positive integers. Find the number of one-to-one functions from A to B.
Suppose /(x) = Va.x+b, where a > 0 and a, b are constants in R. What is the inverse function / {x)? Find the parameters m, n in terms of a, b. The inverse function is given by x) = m(x + n)- where, Suppose f(x) = V3.+6. What is the domain off-tx)? Select one: a. (-0,6] b. Other c. [0,00) d. R e. [6,00) Consider the function 3-X 3x-4 if x <1 if 1sx52 if x > 2 Find...
DO NOT COPY OTHER ANSEWERS!!!! 2. (10 points) Let (%)n>o be a simple symmetric random walk. Compute P(Sn-y|S,n-x) for the two cases n > m and n < m
5. Let f(x) = ax2 +bx+c, where a > 0. Prove that the secant method for minimization will terminate in exactly one iteration for any initial points Xo, X1, provided that x1 + xo: 6. Consider the sequence {x(k)} given by i. Write down the value of the limit of {x(k)}. ii. Find the order of convergence of {x(k)}. 7. Consider the function f(x) = x4 – 14x3 + 60x2 – 70x in the interval (0, 2). Use the bisection...