Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that K...

Let K = { s+t * 2^(1/2), such that s, t are Rational}. Show that K is a Field

Expert Solution

Colby Messinger answered 2 years ago

Let S = {2 k : k ∈ Z}. Let R be a relation defined on...

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (?...

6. Let t be a positive integer. Show that 1? + 2? + ⋯ + (? − 1)? + ?? is ?(??+1). 7. Arrange the functions ?10, 10?, ? log ? , (log ?)3, ?5 + ?3 + ?2, and ?! in a list so that each function is big-O of the next function. 8. Give a big-O estimate for the function ?(?)=(?3 +?2log?)(log?+1)+(5log?+10)(?3 +1). For the function g in your estimate f(n) is O(g(n)), use a simple function g...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is...

Let k ≥ 2. Use that R (the real numbers) is complete to show R^k is complete.

(a) Show that the lines r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s)...

(a) Show that the lines r 1 (t) = (2,1,−4) + t(−1,1,−1) and r 2 (s) = (1,0,0) + s(0,1,−2) are skew. (b) The two lines in (a) lie in parallel planes. Find equations for these two planes. Express your answer in the form ax+by+cz +d = 0. [Hint: The two planes will share a normal vector n. How would one find n?] would one find n?]

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then,...

Let S_k(n) = 1^k + 2^k + ··· + n^k for n, k ≥ 1. Then, S_4(n) is given by S_4(n)= n(n+1)(2n+1)(3n^2 +3n−1)/ 30 Prove by mathematical induction.

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G...

1) a) Let k ≥ 2 and let G be a k-regular bipartite graph. Prove that G has no cut-edge. (Hint: Use the bipartite version of handshaking.) b) Construct a simple, connected, nonbipartite 3-regular graph with a cut-edge. (This shows that the condition “bipartite” really is necessary in (a).) 2) Let F_n be a fan graph and Let a_n = τ(F_n) where τ(F_n) is the number of spanning trees in F_n. Use deletion/contraction to prove that a_n = 3a_n-1 - a_n-2...

Let S={1,2,3,6} and define the relation ~ on S2 by (m,n) ~ (k,l) for m+l=n+k Show...

Let S={1,2,3,6} and define the relation ~ on S2 by (m,n) ~ (k,l) for m+l=n+k Show that it is an equivalent relation Find the number of different equivalent classes and write all of them

Let T and S be linear transformations of a vector space V, and TS=ST (a) Show...

Let T and S be linear transformations of a vector space V, and TS=ST (a) Show that T preserves the generalized eigenspace and eigenspace of S. (b) Suppose V is a vector space on R and dimV = 4. S has a minimal polynomial of (t-2)2 (t-3)2?. What is the jordan canonical form of S. (c) Show that the characteristic polynomial of T has at most 2 distinct roots and splits completely.

(1) Let G be a group and H, K be subgroups of G. (a) Show that...

(1) Let G be a group and H, K be subgroups of G. (a) Show that if H is a normal subgroup, then HK = {xy|x ? H, y ? K} is a subgroup of G. (b) Show that if H and K are both normal subgroups, then HK is also a normal subgroup. (c) Give an example of subgroups H and K such that HK is not a subgroup of G.

Show that Cp = T(∂S/∂T)p and Cv = T(∂S/∂T)V

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