 5.3.1: Consider the following operation: Restrict Pet where PetType = Cat ...
 5.3.2: Consider the following operation: Project Person over (Name, City, ...
 5.3.3: Write the results of the following operation: Project Pet over (Pet...
 5.3.4: Write the results of the following operation: Restrict PetOwner whe...
 5.3.5: A library maintains a database about its books. Information kept on...
 5.3.6: In a relational model of the library database, there is an author r...
 5.3.7: Restrict Author where Country = U. S. giving Results7
 5.3.8: Restrict Writes where Name = Dorothy King giving Results8
 5.3.9: Restrict Book where Publisher = Bellman or Publisher = Swift Key gi...
 5.3.10: Restrict Book where Publisher = Harding and Subject = Art giving Re...
 5.3.11: Project Author over Name giving Results11
 5.3.12: Project Author over (Name, Country) giving Results12
 5.3.13: Project Book over (Publisher, Subject) giving Results13
 5.3.14: Project Book over (ISBN, Title, Subject) giving Results14
 5.3.15: Join Book and Writes over ISBN giving Results15
 5.3.16: Join Author and Writes over Name giving Results16
 5.3.17: For Exercises 1723, using the relation tables given before Exercise...
 5.3.18: For Exercises 1723, using the relation tables given before Exercise...
 5.3.19: For Exercises 1723, using the relation tables given before Exercise...
 5.3.20: For Exercises 1723, using the relation tables given before Exercise...
 5.3.21: For Exercises 1723, using the relation tables given before Exercise...
 5.3.22: For Exercises 1723, using the relation tables given before Exercise...
 5.3.23: For Exercises 1723, using the relation tables given before Exercise...
 5.3.24: If the tuple Fleur, Suzanne NULL gets added to the Author table, wr...
 5.3.25: Suppose a join operation over some attribute is to be done on two t...
 5.3.26: One rule of thumb about good database design is one fact, one place...
 5.3.27: a. Write an SQL query to give the authors name, the title and ISBN ...
 5.3.28: What database integrity errors would be caused by attempting each o...
 5.3.29: A corporation sponsors a yearly campaign to solicit monetary contri...
 5.3.30: Draw an ER diagram based on Exercise 29.
 5.3.31: A universal relation contains all the data values in one relation. ...
 5.3.32: a. If Mary Black moves from the Accounting Department to the Sales ...
 5.3.33: a. If Mary Black moves from the Accounting Department to the Sales ...
 5.3.34: Write an SQL query to give the contribution ID, pay date, and payme...
 5.3.35: Write an SQL query to give the first and last names and payment amo...
 5.3.36: Write an SQL query to reproduce the universal relation of Exercise ...
Solutions for Chapter 5.3: Relations and Databases
Full solutions for Mathematical Structures for Computer Science  7th Edition
ISBN: 9781429215107
Solutions for Chapter 5.3: Relations and Databases
Get Full SolutionsSince 36 problems in chapter 5.3: Relations and Databases have been answered, more than 4213 students have viewed full stepbystep solutions from this chapter. Chapter 5.3: Relations and Databases includes 36 full stepbystep solutions. This textbook survival guide was created for the textbook: Mathematical Structures for Computer Science, edition: 7. Mathematical Structures for Computer Science was written by Patricia and is associated to the ISBN: 9781429215107. This expansive textbook survival guide covers the following chapters and their solutions.

Basis for V.
Independent vectors VI, ... , v d whose linear combinations give each vector in V as v = CIVI + ... + CdVd. V has many bases, each basis gives unique c's. A vector space has many bases!

Cofactor Cij.
Remove row i and column j; multiply the determinant by (I)i + j •

Column picture of Ax = b.
The vector b becomes a combination of the columns of A. The system is solvable only when b is in the column space C (A).

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.

Distributive Law
A(B + C) = AB + AC. Add then multiply, or mUltiply then add.

Free columns of A.
Columns without pivots; these are combinations of earlier columns.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Linear combination cv + d w or L C jV j.
Vector addition and scalar multiplication.

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

Nilpotent matrix N.
Some power of N is the zero matrix, N k = o. The only eigenvalue is A = 0 (repeated n times). Examples: triangular matrices with zero diagonal.

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

Orthonormal vectors q 1 , ... , q n·
Dot products are q T q j = 0 if i =1= j and q T q i = 1. The matrix Q with these orthonormal columns has Q T Q = I. If m = n then Q T = Q 1 and q 1 ' ... , q n is an orthonormal basis for Rn : every v = L (v T q j )q j •

Particular solution x p.
Any solution to Ax = b; often x p has free variables = o.

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.

Polar decomposition A = Q H.
Orthogonal Q times positive (semi)definite H.

Spanning set.
Combinations of VI, ... ,Vm fill the space. The columns of A span C (A)!

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

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.
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