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Topics in Finite Mathematics

by: Zechariah Hilpert

Topics in Finite Mathematics MATH 210

Zechariah Hilpert
GPA 3.8


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This 3 page Class Notes was uploaded by Zechariah Hilpert on Thursday September 17, 2015. The Class Notes belongs to MATH 210 at University of Wisconsin - Madison taught by Staff in Fall. Since its upload, it has received 19 views. For similar materials see /class/205286/math-210-university-of-wisconsin-madison in Mathematics (M) at University of Wisconsin - Madison.

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Date Created: 09/17/15
tPF39ON t 539 5090 HH mm H F1 HH 5000 MICK MHQ Math 210 Chapter 1 Formulas Set a collection of elements nS number of elements in the set S I E A means that 1 belongs to the set A7 it is an element of A z A means I is not an element of A Union XUY226Cor2 Y lntersection X N Y 2 2 E C and 2 E Y Empty set g Universal set U the set of all the elements we are considering Cartesian product XgtltY z E Xy E Y We have nXgtltY nX nY7 nX1 gtlt X2 gtlt gtlt Xk nX1 nX2 X is a subset of Y X C Y if every element of X is an element of Y X and Y are disjoint if X N Y g Complement of the set X X 2 E U 2 X7 nX nU 7 De Morgan7s Laws A U B A N B 7 A N B A U B These also hold for more sets7 for example AUBUC A B C 7 A B C A UB UC Distributive Laws A N B U C A B U A C7 AUB C AUB AUC Pairwise disjoint sets every two sets are disjoint Partition of a set X a collection of pairwise disjoint sets such that their union is X If X17X27 Xk is a partition of X7 then nX nX1 nX2 nA U B nA nB 7 nA N B nAUBUC nAnBnC7nA B7nA C7nB C nA B C ABzzeAz BA B Example B7 A B is a partition of A U B Sample space 5 consists of all possible outcomes of an experiment 23 Multiplication Principle Consider an experiment of k stagesi If the rst stage has n1 possible results the second has n2 resultsm the kth stage has nk results then there are n1 n2 nk elements in the sample space of the experiment 24 If S is a set With n elements then the number of subsets of S is 2 2 2 2 2 multiply 2 With itself n times 25 Warning People tend to forget the empty set H E0 H 01 533 F1 9 50 53 H E0 Chapter 6 Matrix Algebra and Applications A matrix is a rectangular array If it has m rows and n columns we say that it is an m X n matrix A row matrix is one that consists of only one row A column matrix is one that consists of only one column If A aij and A bij are two m X n matrices we say that they are equal if aij by for all i and j If A and B are two matrices as before then their sum is the matrix AB aij bijl lf 5 is a number and A is as before then multiplying c and A gives the matrix CA calj If U is a row nvector and X is a column nvector we de ne the product UX to be UX ulzl ugzg unzni If A aij is an m X n matrix and B bij is an n X k matrix then we can multiply A and B and the matrix product AB is de ned to be the m X k matrix AB clj where Cij aiibij ai2b2j ainbnjA The n X n matrix that has ls in the diagonal and 0s everywhere else is called the n X n identity matrix and denoted In or just I For every matrix A and the identity matrices I for which the products AI and IA are de ned we have AI A and IA A Inverse of a matrix An n X n matrix A is invertible if there is an n X n matrix A such that AB BA I The matrix B with this property is called the inverse of A and denoted A li Note To show that B is the inverse of A it suf ces to show one of the relations AB I or BA I To nd if a matrix A is invertible and calculate the inverse you must a Form the matrix AlIli b Use matrix operations to try and transorm this matrix to one of the form IlB for some matrix c If can be transformed in such a way then A is invertible and A 1 B d If it cannot be transformed to the form IlB then A is not invertible It is not invertible if and only if one of the matrices we get from has a row with only zeros to the left of the vertical line


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