- 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 three-round 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 do-main 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
11
2
This expansive textbook survival guide covers the following chapters and their solutions. Chapter 1.SE includes 46 full step-by-step 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 step-by-step solutions from this chapter. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.
Key Math Terms and definitions covered in this textbook
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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.
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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.
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Complex conjugate
z = a - ib for any complex number z = a + ib. Then zz = Iz12.
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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.
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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.
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Diagonal matrix D.
dij = 0 if i #- j. Block-diagonal: zero outside square blocks Du.
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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).
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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.
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Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.
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Gram-Schmidt 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.
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Hessenberg matrix H.
Triangular matrix with one extra nonzero adjacent diagonal.
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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.
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Plane (or hyperplane) in Rn.
Vectors x with aT x = O. Plane is perpendicular to a =1= O.
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Row picture of Ax = b.
Each equation gives a plane in Rn; the planes intersect at x.
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Spectral Theorem A = QAQT.
Real symmetric A has real A'S and orthonormal q's.
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Toeplitz matrix.
Constant down each diagonal = time-invariant (shift-invariant) filter.
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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·
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Tridiagonal matrix T: tij = 0 if Ii - j I > 1.
T- 1 has rank 1 above and below diagonal.
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Unitary matrix UH = U T = U-I.
Orthonormal columns (complex analog of Q).
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Wavelets Wjk(t).
Stretch and shift the time axis to create Wjk(t) = woo(2j t - k).