 Chapter 3.1: Let R and S be relations on a set A. Suppose we are told that R S. ...
 Chapter 3.2: Let R be the relation on the set of all human beings (not just thos...
 Chapter 3.3: Which of the following relations R defined on the set of all human ...
 Chapter 3.4: Let A D f1; 2; 3; 4g. How many different relations on A are there?
 Chapter 3.5: Let x and y be integers. Suppose x y .mod 10/ and x y .mod 11/. Do ...
 Chapter 3.6: Let R D f.x; y/ W x; y 2 Z and jxj D jyjg. a. Prove that R is an eq...
 Chapter 3.7: Let A D f1; 2; 3g, B D f4; 5g, and R D .AA/[.B B/. Note that R is a...
 Chapter 3.8: Let A D f1; 2; 3; 4; 5g and define an equivalence relation R on 2 A...
 Chapter 3.9: Let P D fN; Z; Pg be a partition of the integers, Z defined by N D ...
 Chapter 3.10: Ten married couples are seated around a large circular table. In ho...
 Chapter 3.11: The letters in the word ELECTRICITY are scrambled to make two, poss...
 Chapter 3.12: Two children are playing tictactoe. In how many ways can the firs...
 Chapter 3.13: There are 21 students in a chemistry class. The students must pair ...
 Chapter 3.14: Let A D f1; 2; 3; : : : ; 100g. How many 10element subsets of A co...
 Chapter 3.15: The expression .x C 2/50 is expanded. What is the coefficient of x 17?
 Chapter 3.16: . Let n be a positive integer. Simplify the following expression: n...
 Chapter 3.17: In a school of 200 children, 15 students are chosen to be on the sc...
 Chapter 3.18: Let n and k be positive integers with k C 2 n. Prove the identity n...
 Chapter 3.19: Let n and k be positive integers. Consider this equation: x1 C x2 C...
 Chapter 3.20: A pizza restaurant features ten different kinds of toppings. When y...
 Chapter 3.21: Let n be a positive integer. How many multisets can be made using t...
 Chapter 3.22: The squares of a 4 4 checkerboard are colored black or white. Use i...
Solutions for Chapter Chapter 3: Counting and Relations
Full solutions for Mathematics: A Discrete Introduction  3rd Edition
ISBN: 9780840049421
Solutions for Chapter Chapter 3: Counting and Relations
Get Full SolutionsSince 22 problems in chapter Chapter 3: Counting and Relations have been answered, more than 9270 students have viewed full stepbystep solutions from this chapter. Mathematics: A Discrete Introduction was written by and is associated to the ISBN: 9780840049421. Chapter Chapter 3: Counting and Relations includes 22 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematics: A Discrete Introduction, edition: 3. This expansive textbook survival guide covers the following chapters and their solutions.

Cyclic shift
S. Permutation with S21 = 1, S32 = 1, ... , finally SIn = 1. Its eigenvalues are the nth roots e2lrik/n of 1; eigenvectors are columns of the Fourier matrix F.

Diagonalizable matrix A.
Must have n independent eigenvectors (in the columns of S; automatic with n different eigenvalues). Then SI AS = A = eigenvalue matrix.

Ellipse (or ellipsoid) x T Ax = 1.
A must be positive definite; the axes of the ellipse are eigenvectors of A, with lengths 1/.JI. (For IIx II = 1 the vectors y = Ax lie on the ellipse IIA1 yll2 = Y T(AAT)1 Y = 1 displayed by eigshow; axis lengths ad

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.

Identity matrix I (or In).
Diagonal entries = 1, offdiagonal entries = 0.

Linearly dependent VI, ... , Vn.
A combination other than all Ci = 0 gives L Ci Vi = O.

Normal equation AT Ax = ATb.
Gives the least squares solution to Ax = b if A has full rank n (independent columns). The equation says that (columns of A)ยท(b  Ax) = o.

Nullspace matrix N.
The columns of N are the n  r special solutions to As = O.

Partial pivoting.
In each column, choose the largest available pivot to control roundoff; all multipliers have leij I < 1. See condition number.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank r (A)
= number of pivots = dimension of column space = dimension of row space.

Simplex method for linear programming.
The minimum cost vector x * is found by moving from comer to lower cost comer along the edges of the feasible set (where the constraints Ax = b and x > 0 are satisfied). Minimum cost at a comer!

Spectrum of A = the set of eigenvalues {A I, ... , An}.
Spectral radius = max of IAi I.

Standard basis for Rn.
Columns of n by n identity matrix (written i ,j ,k in R3).

Stiffness matrix
If x gives the movements of the nodes, K x gives the internal forces. K = ATe A where C has spring constants from Hooke's Law and Ax = stretching.

Symmetric factorizations A = LDLT and A = QAQT.
Signs in A = signs in D.

Symmetric matrix A.
The transpose is AT = A, and aU = a ji. AI is also symmetric.

Toeplitz matrix.
Constant down each diagonal = timeinvariant (shiftinvariant) filter.

Trace of A
= sum of diagonal entries = sum of eigenvalues of A. Tr AB = Tr BA.