consider the set A={ (n-m)(n+m) | n, m m ∈N*, n>m} and B= {2^1+2^2+...+2^k | k∈N*}.
Identify A∩ B
Answer
question
consider the set A={ (n-m)(n+m) | n, m m ∈N*, n>m} and B= {2^1+2^2+...+2^k | k∈N*}....
Consider the set of all functions from {1, 2, ..., m} to {1, 2, ..., n}, where n > m. If a function is chosen from this set at random, what is the probability that it will be strictly increasing? (A) (n)/m”. (B) (%)/nm. () (min-1)/m". (D) (matema!)/n".
Let A be a set with m elements and B be a set with n elements in it. -When is it possible to have a k-to-1 function f such that f : A → B? -Count the number of k-to-1 functions f such that f : A → B
Therom 1.8.2
n choose k = (n choose n-k)
n choose k = (n-1 choose K) + (n-1 choose K-1)
2n = summation of (n choose i )
please use the induction method
(a) (10 pts) Show that the following equality holds: n +1 + 2 Hint: If you proceed by induction, you might want to use Theorem 1.8.2. If you search for a combinatorial proof, consider the set X - (i,j, k): 0 S i,j< k< n) (b) (10...
1. Consider the function h:Z+ +Z+ defined by h(n) = l{k e Z+ : k|n}l. The bars around the set mean that we are taking the size of the set. Thus h(n) is the number of positive divisors of n. (a) Make a table of values for h(n) for 1 sn < 10. Write one or two sentences describing how you found the values in the table. (b) Find the value of h(90). Explain how you found your answer. (c)...
Consider the function 0 : 2+ + 2+ $(n) = number of integers k, 1 <k <n that are relatively prime with n (that is such that (k, n)-1) If n is a prime number º(n) = n n-1 O 1
1. Consider the system shown. Assume B-3 N-s/m and K-7 N/m. Negligible Mass a) Find the transfer function, H(s)-X(s)Fa(s) b) Using the transfer function, find the unit step response and the unit impulse response. c) Using the transfer function, find the steady-state response when fa(t) 2 sin (4t) d) Find the free response (zero-input response) assuming x(0) 2 m.
1. Suppose N is a set with n elements and M is a set with m elements. a. If n <m, how many one-to-one functions are there from N to M? b. If n > m, how many onto functions are there from N to M?
2.
Exercise 2. Consider the sequence (xn)n≥1 defined by xn = Xn k=1
cos(k) k + n2 = cos(1) 1 + n2 + cos(2) 2 + n2 + · · · + cos(n) n +
n2 . (a) Use the triangle inequality to prove that |xn| ≤ n 1 + n2
for all n ≥ 1. (b) Use (a) and the -definition of limit to show
that limn→∞ xn = 0.
Exercise 2. Consider the sequence (In)n> defined by cos(k)...
of S, namely, S- S = {m -n: m, n E S,m >n, is a set of recurrence. Hint: consider the proof of PRT. (b) Let R be a set of integers that contains arbitrarily long arithmetic progressions of the form {n, 2n, ..., kn}. Show that R is a set of recurrence.
Consider N and the set S={x∈{0,...N-1}:gcd(x,N)=1} where k=|S| For a∈S, we define T={ax(modN):x∈S}. what is |T|? Answer may include N and k.