(e) ((SNT)US) (f) U-S (called the complement of S) Given that the set S contains n elements, show that S has 2" subsets. Let S stand for the number of elements in S. Show the validity of |SUT|=|S|+|T| − |SnT|. Show that the following are equivalent. (a) SCT (b) SnT=S (c) SUT=T

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LUP Decomposition

LU decomposition stands for Lower-Upper decomposition. LU decomposition is the factorization of the lower triangular matrix and an upper triangular matrix. It is very useful in linear algebra and numerical analysis. A permutation matrix is usually included in the product. Gaussian elimination in a matrix form is known as LU decomposition, When inverting a matrix or evaluating its determinant, this is a crucial step. Tadeusz Banachiewicz, a Polish mathematician, first proposed the LU decomposition in 1938.