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

CHAP. 15] BOOLEAN ALGEBRA 405 Fig. 15-39 15.64. Find the output sequenceYfor an OR gate with inputsA,B,C(or equivalently forY=A+ ...

406 BOOLEAN ALGEBRA [CHAP. 15 15.71. Find all possible minimal sums for each Boolean expressionEgiven by the Karnaugh maps in Fi ...

CHAP. 15] BOOLEAN ALGEBRA 407 Answers to Supplementary Problems 15.43. (a)a+a′b=a+b. (b)a· 0 +a· 1 =a. (c)ab+bc=(a+c)b. 15.45. ( ...

408 BOOLEAN ALGEBRA [CHAP. 15 15.63. (a)Y=100001;(b)Y=00111000; (c)Y=00110000. 15.64. (a)Y=100111;(b)Y=10100111; (c)Y=11111101. ...

APPENDIX A Vectors and Matrices A.1Introduction Data is frequently arranged inarrays, that is, sets whose elements are indexed b ...

410 VECTORS AND MATRICES [APP. A Thesumofuandv, writtenu+v, is the vector obtained by adding corresponding components fromuandv; ...

APP. A] VECTORS AND MATRICES 411 Themhorizontal lists of numbers are called therowsofA, and thenvertical lists of numbers itscol ...

412 VECTORS AND MATRICES [APP. A EXAMPLE A.4 LetA= [ 1 − 23 045 ] andB= [ 468 1 − 3 − 7 ] . Then A+B= [ 1 + 4 − 2 + 63 + 8 0 + 1 ...

APP. A] VECTORS AND MATRICES 413 (b)[ 6 ,− 1 , 8 , 3 ] ⎡ ⎢ ⎢ ⎣ 4 − 9 − 2 5 ⎤ ⎥ ⎥ ⎦=^24 +^9 −^16 +^15 =^32 We are now ready to de ...

414 VECTORS AND MATRICES [APP. A Matrix multiplication does, however, satisfy the following properties: Theorem A.2: LetA,B,Cbe ...

APP. A] VECTORS AND MATRICES 415 Both are square matrices. The first is of order 3, and its diagonal consists of the elements 1, ...

416 VECTORS AND MATRICES [APP. A Then AB= [ 6 − 5 − 10 + 10 3 − 3 − 5 + 6 ] = [ 10 01 ] and BA= [ 6 − 515 − 15 − 2 + 2 − 5 + 6 ] ...

APP. A] VECTORS AND MATRICES 417 (b) ∣ ∣ ∣ ∣ ∣ ∣ 21 3 46 − 1 51 0 ∣ ∣ ∣ ∣ ∣ ∣ = 2 ( 6 )( 0 )+ 1 (− 1 )( 5 )+ 3 ( 1 )( 4 )− 5 ( 6 ...

418 VECTORS AND MATRICES [APP. A A.10Elementary Row Operations, Gaussian Elimination (Optional) This section discusses the Gauss ...

APP. A] VECTORS AND MATRICES 419 Gaussian Elimination in Matrix Form Consider any matrixA. Two algorithms, Algorithms A-1 and A- ...

420 VECTORS AND MATRICES [APP. A Fig. A-5 The final Steprin Algorithm A-2 in Fig. A-6 changes the first pivot to 1. Fig. A-6 Alg ...

APP. A] VECTORS AND MATRICES 421 Part A (Reduction): Reduce the augmented matrixMto echelon form. If a row of the form( 0 , 0 ,. ...

422 VECTORS AND MATRICES [APP. A EXAMPLE A.12 Find the inverse ofA= ⎡ ⎣ 102 2 − 13 418 ⎤ ⎦. Form the matrixM=(A, I )and reduceMt ...

APP. A] VECTORS AND MATRICES 423 SolvedProblems VECTORS A.1. Letu=( 2 ,− 7 , 1 ),v=(− 3 , 0 , 4 ), andw=( 0 , 5 ,− 8 ). Find: (a ...

424 VECTORS AND MATRICES [APP. A A.7. Findx,y,z,t, where 3 [ xy zt ] = [ x 6 − 12 t ] + [ 4 x+y z+t 3 ] . First write each side ...