Whether the function f ( n , m ) = 2 n + 3 m ; f : N × N → N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Question

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Answer 1

Question

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

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Problem 11 (a) A tank is discharging water through an orifice at a depth of T meter below the surface of the water whose area is A m². The following are the values of a for the corresponding values of A: A 1.257 1.390 x 1.50 1.65 1.520 1.650 1.809 1.962 2.123 2.295 2.462|2.650 1.80 1.95 2.10 2.25 2.40 2.55 2.70 2.85 Using the formula -3.0 (0.018)T = dx. calculate T, the time in seconds for the level of the water to drop from 3.0 m to 1.5 m above the orifice. (b) The velocity of a train which starts from rest is given by the fol- lowing table, the time being reckoned in minutes from the start and the speed in km/hour: | † (minutes) |2|4 6 8 10 12 14 16 18 20 v (km/hr) 16 28.8 40 46.4 51.2 32.0 17.6 8 3.2 0 Estimate approximately the total distance ran in 20 minutes.

- Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p − 1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., p-1 2 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). 23 32 how come? The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. The set T is the subset of these residues exceeding So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1.

Let n = 7, let p = 23 and let S be the set of least positive residues mod p of the first (p-1)/2 multiple of n, i.e. n mod p, 2n mod p, ..., 2 p-1 -n mod p. Let T be the subset of S consisting of those residues which exceed p/2. Find the set T, and hence compute the Legendre symbol (7|23). The first 11 multiples of 7 reduced mod 23 are 7, 14, 21, 5, 12, 19, 3, 10, 17, 1, 8. 23 The set T is the subset of these residues exceeding 2° So T = {12, 14, 17, 19, 21}. By Gauss' lemma (Apostol Theorem 9.6), (7|23) = (−1)|T| = (−1)5 = −1. how come?

Answer 2

Question

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

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Answer 3

Question

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

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Answer 4

Question

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

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Answer 5

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 6

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 7

Chapter 3.1, Problem 22E

a)

To determine

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 8

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 9

b)

To determine

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 10

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 11

c)

To determine

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 12

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 13

d)

To determine

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 14

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 15

e)

To determine

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 16

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 17

f)

To determine

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Answer 18

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Answer 19

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Answer 20

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Answer 21

Ch. 3.1 - True/False Questions A function from a set A to a...Ch. 3.1 - Prob. 2TFQ Ch. 3.1 - Prob. 3TFQ Ch. 3.1 - Prob. 4TFQ Ch. 3.1 - Prob. 5TFQ Ch. 3.1 - True/False Questions Define f:ZZ by f(x)=x+2. Then...Ch. 3.1 - Prob. 7TFQ Ch. 3.1 - Prob. 8TFQ Ch. 3.1 - Prob. 9TFQ Ch. 3.1 - Prob. 10TFQ

Whether the function f ( n , m ) = 2 n + 3 m ; f : N × N → N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Solution Summary: The author explains that f is not one-to-one as some values in range have more than one pre-image.

Whether the function f(n,m)=2n+3m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Whether the function f(n,m)=2n+3m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Whether the function f(n,m)=14n+22m;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Whether the function f(n,m)=89n+246m;f:Z×Z→Z is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Whether the function f(n,m)=n2+m2+1;f:Z×Z→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

Whether the function f(n,m)=⌊nm⌋+1;f:N×N→N is one-to-one and/or onto. Also give a proof or exhibit a counterexample to justify the answer.

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Chapter 3 Solutions