Let x, y e R" for n e N, writing x-(n, ,%), similarly for y. The Euclidean Metric of $2.2 is often called the 12 metric and written |x - y l2 for x,y e R. Show that the following three similar relations are also metrics: (a) the tancab, or 11 metric: lx-wi : = Σ Iri-Vil (b) the marirnum, or lo-metric. Ilx-ylloo:=max(zi-yil c) the comparison, or 10-metric. Ix_ylo rn (ri-Vi) where δ(t)- if t = 0
please help with 6a b and C 6. Prove by strong induction: Any amount of past be made using S 7 and 13 cent stamps. (Fill in the blank with the smallest number that makes the statement true). Let fib(n) denote the nth Fibonacci number, so fib(0) - 1, fib(1) - 1, fib(2) -1, fib(3) = 2, fib(4) – 3 and so on. Prove by induction that 3 divides fib(4n) for any nonnegative integer n. Hint for the inductive step:...
Prove by strong induction that any nonzero natural number can be written as a sum of distinct powers of 2.
Recall that for any integers x and y, we say that x is divisible by y if and only if there exists an integer k such that x=ky. Prove by induction the following claim: n^3 + 2n is divisible by 3. ( n^3 =n*n*n)
Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove that d is a metric on R. (2) Letting xnn, prove that {xnJnE is a Cauchy sequence with no limit in R (Note that {xn)nen is NOT Cauchy under the Euclidean metric and that all Cauchy sequences in the Euclidean metric have a limit in R.) Problem 2: For any x, y e R let d(x,y):-arctan(y) - arctan(x). Do the following: (1) Prove...
(b) A Y-connected induction motor. The equivalent circuit parameters are R X, X, M PF&W P. misc core (lumped with rotational losses) [14 points) For a slip Sind (1) The line current I _ - [3 points] (2) The stator power factor. [1 point] (3) The rotor power factor. [1 point] (4) The rotor frequency. [1 point] (5) The stator copper losses Pscz - [1 point] (6) The air-gap power PAG - [1 point] (7) The power converted from electrical...
11. Prove Bonferroni's ineqyuality r (n A) PLA)-(1) by induction n 1) by induction.
Define a relation R on N x N by R = {(x,y) | x ε N, y ε N and x+y is even} Prove or disprove: R is an equivalence relation.
5. Use induction to prove the following for x,y EQ and n, mEN. (c) xn = 0 iff x = 0 (d) If x 〉 y 〉 0, then xn 〉 yn 〉 0
5. Partitions For each n e Z, let T={(x, y) + R n<I- g < n+1}. Is T = {T, n € Z} a partition of R?? Justify your answer using the definition.