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

CHAP. 11] PROPERTIES OF THE INTEGERS 299

SupplementaryProblems

ORDER AND INEQUALITIES, ABSOLUTE VALUE

11.61. Insert the correct symbol,<, >,or=, between each pair of integers:

(a)2 ___− 6 ; (c)−7 ___3; (e) 23 ___11; (g)−2 ___− 7 ;

(b)−3 ___− 5 ; (d)−8 ___− 1 ; (f) 23 ___− 9 ; (h)4 ___− 9.

11.62. Evaluate: (a)| 3 − 7 |,|− 3 + 7 |,|− 3 − 7 |; (b)| 2 − 5 |+| 3 + 7 |,| 1 − 4 |−| 2 − 9 |;
(c)| 5 − 9 |+| 2 − 3 |,|− 6 − 2 |−| 2 − 6 |.

11.63. Find the distancedbetween each pair of integers: (a) 2 and−5; (b)−6 and 3; (c) 2 and 8; (d)−7 and−1;
(e) 3 and−3; (f)−7 and−9.

11.64. Find all integersnsuch that: (a) 3< 2 n− 4 <10; (b) 1< 6 − 3 n<13.

11.65. ProveProposition11.1: (i)a≤a, for any integera; (ii) Ifa≤bandb≤a, thena=b.

11.66. Prove Proposition 11.2: For any integersaandb, exactly one of the following holds:a<b,a=b,ora>b.

11.67. Prove: (a) 2ab≤a^2 +b^2 ; (b)ab+ac+bc≤a^2 +b^2 +c^2.

11.68. Proposition 11.4: (i)|a|≥0, and|a|=0iffa=0; (ii)−|a|≤a≤|a|; (iii)||a|−|b||≤|a±b|.

11.69. Show thata−xb≥0ifb=0, andx=−|a|b.

MATHEMATICAL INDUCTION,WELL-ORDERING PRINCIPLE

11.70. Prove the proposition that the sum of the firstneven positive integers isn(n+ 1 ); that is,

P (n): 2 + 4 + 6 +···+ 2 n−=n(n+ 1 )

11.71. Prove that the sum of the firstncubes is equal to the square of the sum of the firstnpositive integers:

P (n): 13 + 23 + 33 +···+n^3 =( 1 + 2 +···+n)^2

11.72. Prove: 1+ 4 + 7 +···+( 3 n− 2 )=n( 3 n− 1 )/ 2

11.73. Prove: (a)anam=an+m; (b)(an)m=anm; (c)(ab)n=anbn

11.74. Prove: 11 · 2 + 21 · 3 + 31 · 4 +···+n(n^1 + 1 )=n+n 1

11.75. Prove: 11 · 3 + 31 · 5 + 51 · 7 +···+( 2 n− 1 )(^12 n+ 1 )= 2 nn+ 1

11.76. Prove:^1

2

1 · 3 +

22

3 · 5 +

32

5 · 7 +···+

n^2

( 2 n− 1 )( 2 n+ 1 )=

n(n+ 1 )

2 ( 2 n+ 1 )

11.77. Prove:xn+^1 −yn+^1 =(x−y)(xn+xn−^1 y+xn−^2 y^2 +···+yn)

11.78. Prove:|P (A)|= 2 nwhere|A|=n. (HereP (A)is the power set of the setAwithnelements.)

DIVISION ALGORITHM

11.79. For each pair of integersaandb, find integersqandrsuch thata=bq+rand 0≤r<|b|:

(a)a=608 andb=−17; (b)a=−278 andb= 12 ; (c)a=−417 andb=− 8.

11.80. Prove each of the following statements:

(a) Any integerais of the form 5k,5k+1, 5k+2, 5k+3, or 5k+4.

(b) One of five consecutive integers is a multiple of 5.

11.81. Prove each of the following statements:

(a) The product of any three consecutive integers is divisible by 6.

(b) The product of any four consecutive integers is divisible by 24.

11.82. Show that each of the following numbers is not rational: (a)

√

3 ; (b)^3

√

2.

11.83. Show that

√

pis not rational, wherepis any prime number.