Set Proof:
1. Prove that if S and T are finite sets with |S| = n and |T| = m, then |S U T| <= (n + m)
2. Prove that finite set S = T if and only if (iff) (S Tc) U (Sc T) =
ILULIITUL 10.37 Theorem. (The Generalized Distributive Laws for Sets of Sets.) Let S be a set and let be a non-empty set of sets. Then: (a) SNU =USNA: AE}. (b) Sund= {SUA:AE). Proof (a) Let = {SNA: AE }. We wish to show that S U = UB. For each 1, we have BESUS iff x S and 2 EU iff xe S and there exists AE such that EA iff there exists AE such that reS and x E...
Let and be two finite measures on . Prove that if and only if the condition implies , for each . Thank you for your explanations. We were unable to transcribe this imageWe were unable to transcribe this image(N, P (N)) μ<<φ 6({n})=0 ({n}) = 0 neN
Let V be a finite-dimensional vector space and let T L(V) be an operator. In this problem you show that there is a nonzero polynomial such that p(T) = 0. (a) What is 0 in this context? A polynomial? A linear map? An element of V? (b) Define by . Prove that is a linear map. (c) Prove that if where V is infinite-dimensional and W is finite-dimensional, then S cannot be injective. (d) Use the preceding parts to prove...
Let n, and let n be a reduced residue. Let r = odd(). Prove that if r = st for positive integers s and t, then old(t) = s. We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Find the image of the set S under the following transformations. 1) S= {(u,v) R2 | 0 u 3, 0 u 2 }, x=2u+3v and y=u-v 2) S= [0,1] x [0,1], x=v and y=u(1+v2) We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
Let A and B be finite sets. The properties of set operations, prove that: notation denotes the complement. Let the universal set be U. Usin (AUB) n (AUBc) = A
3. (20 pts) Let ụ be a finite set, and let S = {Si, S , S,n} be a collection of subsets of U. Given an integer k, we want to know if there is a sub-collection of k sets S' C S whose union covers all the elements of U. That is, S k, and Us es SU. Prove that this problem is NP-complete. 992 m SES, si 3. (20 pts) Let ụ be a finite set, and let...
Define , a finite -group, such that isn't abelian. Let such that , where is abelian. Prove that there are either or such abelian subgroups, and if there are , then the index of in is T We were unable to transcribe this imageT K G:K=P We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageT We were unable...
Define a prime number, a finite group, as a Sylow -subgroup of . Assume there exists a proper subgroup of where , i.e. the normaliser of in is a subgroup of . Prove that isn't normal in . We were unable to transcribe this imageT We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageNG(K) < M We were...
When there is a finite set of N = {1, ... n} as individuals, and there is n>X>=3 where X are possible alternatives. And R is the set of orderings of X and B is the set of binary relations of X. for all x, y in X and R in Rn, define arrovian social welfare function f:Rn -> B as 1) xPy IFF xPiy for at least n-1 individuals 2) xRy IFF not(yPx) where R=f(R). Prove that R=f(R) is...