 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 254793 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.

Augmented matrix [A b].
Ax = b is solvable when b is in the column space of A; then [A b] has the same rank as A. Elimination on [A b] keeps equations correct.

Big formula for n by n determinants.
Det(A) is a sum of n! terms. For each term: Multiply one entry from each row and column of A: rows in order 1, ... , nand column order given by a permutation P. Each of the n! P 's has a + or  sign.

Complex conjugate
z = a  ib for any complex number z = a + ib. Then zz = Iz12.

Covariance matrix:E.
When random variables Xi have mean = average value = 0, their covariances "'£ ij are the averages of XiX j. With means Xi, the matrix :E = mean of (x  x) (x  x) T is positive (semi)definite; :E is diagonal if the Xi are independent.

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.

Diagonal matrix D.
dij = 0 if i # j. Blockdiagonal: zero outside square blocks Du.

Dot product = Inner product x T y = XI Y 1 + ... + Xn Yn.
Complex dot product is x T Y . Perpendicular vectors have x T y = O. (AB)ij = (row i of A)T(column j of B).

Echelon matrix U.
The first nonzero entry (the pivot) in each row comes in a later column than the pivot in the previous row. All zero rows come last.

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

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.

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

Independent vectors VI, .. " vk.
No combination cl VI + ... + qVk = zero vector unless all ci = O. If the v's are the columns of A, the only solution to Ax = 0 is x = o.

Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.

Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.

Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.

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

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Tridiagonal matrix T: tij = 0 if Ii  j I > 1.
T 1 has rank 1 above and below diagonal.

Unitary matrix UH = U T = UI.
Orthonormal columns (complex analog of Q).

Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t  k).