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,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that S possesses two dilferent 3-element subsets, the sums of whose elements are equal. (b) Show that S possesses two disjoint subsets, the sums of whose elements are equal ,n2} with ISI = n. 4. Let n be a positive integer with n > 20, and let S {1, 2, -I with a) Show that...
Let ne N. Show that Zn forn > 2 is not a group under multiplication as defined above. What happens for n = 1?
Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction Let G be a simple graph with 2n, n 2 vertices. Suppose there are at least n2 1 edges. Show that at least one triangle is formed. Hint: Check n 2 first and then use induction
Let ne Nj. Prove that n < 2(6(n)).
8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine all such triangles. b. Find the smallest two such triangles. 8. 2mn, b-m2-n, and c = m2 + n2 be the sides of a a. Let a Pythagorean triangle. Suppose that b -a + 1. Show that (m - n)2 - 2n2 1, and determine...
1. Let {Xn,n2 0 be a Markov Chain with state space S. Show that 20 for any n,m-1 and JAn+m , . . . , İn+1,in-1, . . . , io є s.
Make this program using MATLAB AND SHOW THE DETAIL WORKING n2 if n 10 2. Let f(n)3n2 2 if 10 n < 30 4n if n 2 30 Create an m-fhle with a program to find the sum 50 Σ f(n). n=1 Hint: You may find the following commands to be useful: for elseif else end if When you are done, run your program and write the value of the sum here: 50 f(n) Even if you do not manage...
Problem 4. Show that for all integers n, n2 mod 3- 1 n2 mod 3- 0 or (i.e. there exists an integer k such that n2 3k or n2 3k +1). mp
2. (40 pts) Let fn: RR be given by sin(n) In(x) = n2 NEN. 2a. (10 pts) Show that the series 2n=1 fn converges uniformly on R. 2b. (10 pts) Show that the function f: RR, f (x) = sin (nx) n2 n=1 is continuous on R. 2c. (10 pts) Show that f given in 2b) is intergrable and $(z)de = 24 (2n-1) 2d. (10 pts) Let 0 <ö< be given. Show that f given in 2b) is differentiable at...
A gas mixture consists of N2, O2, and Ne, where the mole fraction of N2 is 0.55 and the mole fraction of Ne is 0.25. If the mixture is at STP in a 5.0 L container, how many molecules of O2 are present?