Binomial Coefficients : The coefficient of a^10 b^4 and the coefficient of a^6 b^8
In the expansion (a+b)^14 find: a) The coefficient of a^10 b^4. b) The coefficient of a^6 b^8. --- Definition of Binomial Coefficient: Given a set X and a non-negative integer r, an r-subset of X is a subset A X of cardinality r. We denote the set of r-subsets of X by Pr(X), i.e. Pr(X) = {A X | |A| = r}. We de ...continues
Injections, Surjections and Number of Functions from A to B and B to A
Let A = {1,2,3} and B = {9,11}. a) Determine the number of functions from A to B. b) List the different surjections f : A --> B. How many are there? c) Determine the number of functions from B to A. d) List the different injections g: B —> A. How many are there?.
Binomial Coefficients : Combinations
Four people select a main dish form a menu of 7 items, How many choices are possible: a) if a record is kept of who selected which choice (as a waiter would); b) if who selects which choice is ignored (as a chef would). Analyze this part by the number of different choices made. Definition of Binomial Coefficient: Given ...continues
Solve the System of Equations : Matices, Row Reductions and Echelon Form
Solve the system of equations [ 1 1 0 1 8] [ X1] [ 2] [-1 1 2 -1 0] [ X2] [-1] [-2 0 4 6 2] [ X3] = [ 2] [0 -3 -1 1 4 ] [ 0] [X4] [3 1 2 5-1] [ 1] [X5]
1. Let p be a poistive prime integer. Show that there is no rational number r such that p = r^2. Please see the attached file for the fully formatted problems. ---
Write the following as rational numbers: 0.731; (31 barred) and 12.8697 (8697 barred)
2. Write the following as rational numbers: a) 0.731; (31 barred) b) 12.8697 (8697 barred) Please see the attached file for the fully formatted problems.
Proofs : Denumerable Sets and Cartesian Products
Need help in determining the following proof exercise. (See attached file for full problem description) --- Corollary 14.2.4: If A is a denumerable set then so is A^n for every positive integer n. Proposition 14.2.3: If A and B are denumerable sets then so is their Cartesian product , AX B. ---
Functions, Enumeration Schemes and Bijection
Let f:Z+ --> Z be the function defined by... f(n) = {n/2 n even { -(n-1)/2 n odd — n even; The following table indicates the enumeration scheme behind the definiton of the function. n .......... 7 5 312468... f(n)..... —3 —2 —1 0 1 2 3 4... Show that f is a bijection. [You may assume basic facts about the ...continues
1. Let d; a; b; r, and q be integers. a) Suppose that d|a and d|b. Show that d|(ra + qb). b) Suppose a = qb + r. Show that the set of common divisors of a and b is the set of common divisors of b and r.
Greatest Common Divisors ( GCD )
1. Let a, b be positive integers, and write a = qb + r, where q, r are Elements of Z and 0 (= or)< r < b. Suppose that d = gcd(a, b). a) If r = 0 show that d = b. b) If r > 0 show that d = gcd(b, r). 4. Use Problem 1 to find: a) gcd(100; 3); b) gcd(100; 82).
