Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

APP. A] VECTORS AND MATRICES 425 A.11. Find the transpose of each matrix: A= [ 1 − 23 78 − 9 ] ; B= ⎡ ⎣ 123 245 356 ⎤ ⎦; C=[ 1 , ...

426 VECTORS AND MATRICES [APP. A (c) Computef (A)by first substitutingAforxand 5Ifor the constant term 5 inf(x)= 2 x^3 − 4 x+5: ...

APP. A] VECTORS AND MATRICES 427 (a) Form the matrixM=[A, I]and row reduceMto echelon form: M= ⎡ ⎣ 1 − 22 ... 100 2 − 36 ..... 0 ...

428 VECTORS AND MATRICES [APP. A A.22. Reduce the matrixA= ⎡ ⎣ 1 −23 12 114 − 13 259 − 28 ⎤ ⎦to row canonical form. First reduce ...

APP. A] VECTORS AND MATRICES 429 MISCELLANEOUS PROBLEMS A.24. LetA= ⎡ ⎣ 100 001 110 ⎤ ⎦andB= ⎡ ⎣ 011 100 010 ⎤ ⎦be Boolean matri ...

430 VECTORS AND MATRICES [APP. A A.30. LetA= [ 1 − 12 034 ] ,B= [ 40 − 3 − 1 − 23 ] ,C= ⎡ ⎣ 2 − 301 5 − 1 − 42 − 1003 ⎤ ⎦,D= ⎡ ⎣ ...

APP. A] VECTORS AND MATRICES 431 MISCELLANEOUS PROBLEMS A.45. LetA= [ 12 01 ] .Find: (a)An; (b)A−^1 ; (c) matrixBsuch thatB^2 =A ...

APPENDIX B Algebraic Systems B.1Introduction This Appendix investigates some of the major algebraic systems in mathematics: semi ...

APP. B] ALGEBRAIC SYSTEMS 433 EXAMPLE B.1 Consider the setNof positive integers. (a) Addition (+) and multiplication (×) are ope ...

434 ALGEBRAIC SYSTEMS [APP. B (b) Consider the operation of matrix multiplication on the setMofn-square matrices. One can prove ...

APP. B] ALGEBRAIC SYSTEMS 435 B.3Semigroups LetSbe a nonempty set with an operation. ThenSis called asemigroupif the operation i ...

436 ALGEBRAIC SYSTEMS [APP. B EXAMPLE B.6 (a) LetAandBdenote, respectively, the set of even and odd positive integers. Then (A,× ...

APP. B] ALGEBRAIC SYSTEMS 437 Homomorphism of Semigroups Consider two semigroups (S,∗) and (S′,∗′). A functionf:S→S′is called as ...

438 ALGEBRAIC SYSTEMS [APP. B Fundamental Theorem of Semigroup Homomorphisms Recall that the image of a functionf:S→S′, writtenf ...

APP. B] ALGEBRAIC SYSTEMS 439 EXAMPLE B.10 (a) The nonzero rational numbersQ\{0} form an abelian group under multiplication. The ...

440 ALGEBRAIC SYSTEMS [APP. B MAP(A), PERM(A), and AUT(A) LetAbe a nonempty set. The collection MAP(A) of all functions (mapping ...

APP. B] ALGEBRAIC SYSTEMS 441 EXAMPLE B.11 (a) Consider the permutation groupS 3 of degree 3 which is investigated above. The se ...

442 ALGEBRAIC SYSTEMS [APP. B Cyclic Subgroups LetGbe any group and letabe any element ofG. As usual, we definea^0 =eandan+^1 =a ...

APP. B] ALGEBRAIC SYSTEMS 443 Theorem B.9: Supposef:G→G′is a homomorphism with kernelK. ThenKis a normal subgroup ofG, and the q ...

444 ALGEBRAIC SYSTEMS [APP. B Special Kinds of Rings: Integral Domains and Fields This subsection defines a number of different ...