 1.SE.1E: Let p be the proposition "I will do every exercise in this book" an...
 1.SE.2E: Find the truth table of the compound proposition (p ?q)?(p?¬r).
 1.SE.3E: Show that these compound propositions are tautologies.
 1.SE.4E: Give the converse, the contrapositive, and the inverse of these con...
 1.SE.5E: Given a conditional statement p ? q, find the converse of its inver...
 1.SE.6E: Given a conditional statement p? q, find the inverse of its inverse...
 1.SE.7E: Find a compound proposition involving the propositional variables p...
 1.SE.8E: Show that these statements are inconsistent: "If Sergei lakes the j...
 1.SE.9E: Show that these statements are inconsistent: "If Miranda does not t...
 1.SE.10E: Suppose that in a threeround obligato game, the teacher first give...
 1.SE.11E: Teachers in middle ages supposedly tested the realtime propositiona...
 1.SE.12E: Explain why every obligato game has a winning strategy.Exercises 13...
 1.SE.13E: Suppose that you meet three people Aaron, Bohan, and Crystal. Can y...
 1.SE.14E: Suppose that you meet three people, Anita, Boris, and Carmen. What ...
 1.SE.15E: (Adapted from [Sm78]) Suppose that on an island there are three typ...
 1.SE.16E: Show that if S is a proposition, w here S is the conditional statem...
 1.SE.17E: Show that the argument with premises "The tooth fairy is a real per...
 1.SE.18E: Suppose that the truth value of the proposition pi is T whenever i ...
 1.SE.19E: Model 16 ×16 Sudoku puzzles (with 4×4 blocks) as satisfiability pro...
 1.SE.20E: Let P(x) be the statement "Student x knows calculus" and let Q(y) b...
 1.SE.21E: Let P(m,n) be the statement "m divides n" where the domain for bot...
 1.SE.22E: Find a domain for the quantifiers in such that this statement is true.
 1.SE.23E: Find a domain for the quantifiers in such that this statement is fa...
 1.SE.24E: Use existential and universal quantifiers to express the statement ...
 1.SE.25E: Use existential and universal quantifiers to express the statement ...
 1.SE.26E: The quantifier ?n denotes "there exists exactly n,” so that means t...
 1.SE.27E: Express each of these statements using existential and universal qu...
 1.SE.28E: Let P(x, y) be a propositional function. Show that is a tautology.
 1.SE.29E: Let P(x) and Q(x) be prepositional functions. Show always have the ...
 1.SE.30E: If is true, does it necessarily follow that is true?
 1.SE.31E: If is true, docs it necessarily follow that is true?
 1.SE.32E: Find the negations of these statements.a) If it snows today, then I...
 1.SE.33E: Express this Statement using quantifiers: "Every student in this cl...
 1.SE.34E: Express this statement using quantifiers: 'There is a building on t...
 1.SE.35E: Express the Statement "There is exactly one student in this class w...
 1.SE.36E: Describe a rule of inference that can be used to prove that there a...
 1.SE.37E: Use rules of inference lo show that if the premises and ¬R(a), wher...
 1.SE.38E: Prove that if x2 is irrational, then x is irrational.
 1.SE.39E: Prove that if x is irrational and x ? 0, then is irrational.
 1.SE.40E: Prove that given a nonnegative integer", there is a unique nonnegat...
 1.SE.41E: Prove that there exists an integer m such that m2 > 101000 .Is your...
 1.SE.42E: Prove that there is a positive integer that can be written as the s...
 1.SE.43E: Disprove the statement that every positive integer is the sum of th...
 1.SE.44E: Disprove the statement that every positive integer is the sum of at...
 1.SE.45E: Disprove the statement that every positive integer is the sum of 36...
 1.SE.46E: Assuming the truth of the theorem that stales that is irrational wh...
Solutions for Chapter 1.SE: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 1.SE
Get Full SolutionsThis expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.SE includes 46 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 46 problems in chapter 1.SE have been answered, more than 114988 students have viewed full stepbystep solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Back substitution.
Upper triangular systems are solved in reverse order Xn to Xl.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Commuting matrices AB = BA.
If diagonalizable, they share n eigenvectors.

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.

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Four Fundamental Subspaces C (A), N (A), C (AT), N (AT).
Use AT for complex A.

Fourier matrix F.
Entries Fjk = e21Cijk/n give orthogonal columns FT F = nI. Then y = Fe is the (inverse) Discrete Fourier Transform Y j = L cke21Cijk/n.

Free variable Xi.
Column i has no pivot in elimination. We can give the n  r free variables any values, then Ax = b determines the r pivot variables (if solvable!).

GramSchmidt orthogonalization A = QR.
Independent columns in A, orthonormal columns in Q. Each column q j of Q is a combination of the first j columns of A (and conversely, so R is upper triangular). Convention: diag(R) > o.

Graph G.
Set of n nodes connected pairwise by m edges. A complete graph has all n(n  1)/2 edges between nodes. A tree has only n  1 edges and no closed loops.

Kronecker product (tensor product) A ® B.
Blocks aij B, eigenvalues Ap(A)Aq(B).

Matrix multiplication AB.
The i, j entry of AB is (row i of A)·(column j of B) = L aikbkj. By columns: Column j of AB = A times column j of B. By rows: row i of A multiplies B. Columns times rows: AB = sum of (column k)(row k). All these equivalent definitions come from the rule that A B times x equals A times B x .

Multiplication Ax
= Xl (column 1) + ... + xn(column n) = combination of columns.

Permutation matrix P.
There are n! orders of 1, ... , n. The n! P 's have the rows of I in those orders. P A puts the rows of A in the same order. P is even or odd (det P = 1 or 1) based on the number of row exchanges to reach I.

Projection p = a(aTblaTa) onto the line through a.
P = aaT laTa has rank l.

Semidefinite matrix A.
(Positive) semidefinite: all x T Ax > 0, all A > 0; A = any RT R.

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

Sum V + W of subs paces.
Space of all (v in V) + (w in W). Direct sum: V n W = to}.

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