A Transition to Advanced Mathematics
A Transition to Advanced Mathematics
8th Edition
ISBN: 9781285463261
Author: Douglas Smith, Maurice Eggen, Richard St. Andre
Publisher: Cengage Learning
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Chapter 1.7, Problem 9E

(a)

To determine

To prove: every point inside the circle (x3)2+(y2)2=4 is inside the circle x2+y2=41 .

(a)

Expert Solution
Check Mark

Explanation of Solution

Given information: (x3)2+(y2)2=4

Proof:

On the graph we can see that the claim is correct, let’s prove it formally.

Assume x and y are real numbers such that.

  (x3)2+(y2)24(x3)24(y2)242x322y221x50y4x2+y2=25+16=41

The proof is done.

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  1

(b)

To determine

To prove: if (x,y) is inside the circle (x3)2+(y2)2=4, then x6<3y .

(b)

Expert Solution
Check Mark

Explanation of Solution

Given information: (x3)2+(y2)2=4,

Proof:

On the graph we can see that the claim is correct, let’s prove it formally.

Assume x and y are real numbers such that.

  (x3)2+(y2)24(x3)24(y2)242x322y221x50y45x6103y12x61<03y

The proof is done.

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  2

(c)

To determine

To prove: every point inside the circle (x3)2+(y2)2=4 is inside the circle (x5)2+(y+1)2=25 .

(c)

Expert Solution
Check Mark

Explanation of Solution

Given information: (x3)2+(y2)2=4 .

Proof:

On the graph we can see that the claim is correct, let’s find a counterexample.

Let (x,y) be (2,3,5), then

  (23)2+(352)2=1+152=3254 .

It is in the first circle.

  (25)2+(35+1)2=9+2025=2925>25

It is not in the second circle.

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  3

(d)

To determine

To prove: if (x,y) is inside the circle (x3)2+(y2)2=4 is or the circle (x+1)2+(y3)2=9, then x + y > 3.

(d)

Expert Solution
Check Mark

Explanation of Solution

Given information: (x3)2+(y2)2=4 .

Proof:

On the graph we can see that the claim is correct, let’s find a counterexample.

Let (x,y) be (0,4), then

  (0+1)2+(43)2=1+1=29 .

It is in the second circle.

But

  0+4=4>3

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  4

(e)

To determine

To prove: if (x,y) is inside the circle (x5)2+(y1)2=17 , then y<2 or xy>0 .

(e)

Expert Solution
Check Mark

Explanation of Solution

Given information: (x5)2+(y1)2=17 .

Proof:

On the graph we can see that the claim is correct, let’s prove it formally.

Assume x and y are real numbers such that.

  (x5)2+(y1)217(x5)217(y1)21717x51717y1174.2x54.14.2y14.10.8x9.13.2y5.15x6103y12(y<2y>2)x>0y<2xy>0

The proof is done.

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  5

(f)

To determine

To prove: there is a unique real number x such that for all real numbers y , if y < x then x + y is negative and x − y is positive.

(f)

Expert Solution
Check Mark

Explanation of Solution

Given information: there is a unique real number x such that for all real numbers y, y < x.

Proof:

On the graph we can see that the claim is correct, let’s find a counterexample.

Let (x1,y1) be (2,4) and (x2,y2) be (2,4), then

  4<2 and 4<22+(4)=6<0 and 2+(4)=2<02(4)=2>0 and 2(4)=6>0

  (x,y) exists but it is not unique.

  A Transition to Advanced Mathematics, Chapter 1.7, Problem 9E , additional homework tip  6

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Chapter 1 Solutions

A Transition to Advanced Mathematics

Ch. 1.1 - Give a useful denial of each statement. Assume...Ch. 1.1 - Restore parentheses to these abbreviated...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.1 - Other logical connectives between two propositions...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Prob. 2ECh. 1.2 - What can be said about the truth value of Q when...Ch. 1.2 - Identify the antecedent and the consequent for...Ch. 1.2 - Which of the following conditional sentences are...Ch. 1.2 - Which of the following are true? Assume that x and...Ch. 1.2 - Make truth tables for these propositional forms....Ch. 1.2 - Prove Theorem 1.2.2 by constructing truth tables...Ch. 1.2 - Determine whether each statement qualifies as a...Ch. 1.2 - Prob. 10ECh. 1.2 - Dictionaries indicate that the conditional meaning...Ch. 1.2 - Show that the following pairs of statements are...Ch. 1.2 - Prob. 13ECh. 1.2 - Give, if possible, an example of a false...Ch. 1.2 - Give the converse and contrapositive of each...Ch. 1.2 - Prob. 16ECh. 1.2 - The inverse, or opposite, of the conditional...Ch. 1.3 - Translate the following English sentences into...Ch. 1.3 - For each of the propositions in Exercise 1, write...Ch. 1.3 - Translate these definitions from the Appendix into...Ch. 1.3 - Prob. 4ECh. 1.3 - The sentence “People dislike taxes” might be...Ch. 1.3 - Let T={17},U={6},V={24} , and W={2,3,7,26} . In...Ch. 1.3 - (a) Complete the following proof of Theorem...Ch. 1.3 - Which of the following are true? The universe for...Ch. 1.3 - Give an English translation for each. The universe...Ch. 1.3 - Which of the following are true in the universe of...Ch. 1.3 - Let A(x) be an open sentence with variable x. (a)...Ch. 1.3 - Suppose the polynomials anxn+an1xn1+...+a0 and...Ch. 1.3 - Which of the following are denials of (!x)P(x) ?...Ch. 1.3 - Riddle: What is the English translation of the...Ch. 1.4 - Analyze the logical form of each of the following...Ch. 1.4 - A theorem of linear algebra states that if A andB...Ch. 1.4 - Verify that [(BM)L(ML)]B is a tautology. See the...Ch. 1.4 - These facts have been established at a crime...Ch. 1.4 - Prob. 5ECh. 1.4 - Let a and b be real numbers. Prove that (a)...Ch. 1.4 - Suppose a, b, c, and d are integers. Prove that...Ch. 1.4 - Give two proofs that if n is a natural number,...Ch. 1.4 - Let a, b, and c be integers and x, y, and z be...Ch. 1.4 - Recall that except for degenerate cases, the graph...Ch. 1.4 - Exercises throughout the text with this title ask...Ch. 1.5 - Analyze the logical form of each of the following...Ch. 1.5 - A theorem of linear algebra states that if A andB...Ch. 1.5 - Let x, y, and z be integers. Write a proof by...Ch. 1.5 - Write a proof by contraposition to show that for...Ch. 1.5 - A circle has center (2,4) . (a) Prove that (1,5)...Ch. 1.5 - Suppose a and b are positive integers. Write a...Ch. 1.5 - Prob. 7ECh. 1.5 - Prob. 8ECh. 1.5 - Prove by contradiction that if n is a natural...Ch. 1.5 - Prove that 5 is not a rational number.Ch. 1.5 - Three real numbers, x, y, and z, are chosen...Ch. 1.5 - Assign a grade of A (correct), C (partially...Ch. 1.6 - Prove that (a) there exist integers m and n such...Ch. 1.6 - Prove that for all integers a, b, and c, If...Ch. 1.6 - Prove that if every even natural number greater...Ch. 1.6 - Provide either a proof or a counterexample for...Ch. 1.6 - (a) Prove that the natural number x is prime if...Ch. 1.6 - Prove that (a) for every natural number n, 1n1 ....Ch. 1.6 - Starting at 9 a.m. on Monday, a hiker walked at a...Ch. 1.6 - Show by example that each of the following...Ch. 1.6 - Assign a grade of A (correct), C (partially...Ch. 1.7 - (a) Let a be a negative real number. Prove that if...Ch. 1.7 - Prob. 2ECh. 1.7 - Prove that (a) 5n2+3n+4 is even, for all integers...Ch. 1.7 - Prob. 4ECh. 1.7 - Prove that (a) if x + y is irrational, then either...Ch. 1.7 - Prob. 6ECh. 1.7 - Prob. 7ECh. 1.7 - Prob. 8ECh. 1.7 - Prob. 9ECh. 1.7 - Prob. 10ECh. 1.7 - Assign a grade of A (correct), C (partially...Ch. 1.8 - For each given pair a, b of integers, find the...Ch. 1.8 - Prob. 2ECh. 1.8 - Let a and b be integers, a0 , and ab . Prove that...Ch. 1.8 - Prob. 4ECh. 1.8 - Prob. 5ECh. 1.8 - Prob. 6ECh. 1.8 - Prob. 7ECh. 1.8 - Prob. 8ECh. 1.8 - Prove that for every prime p and for all natural...Ch. 1.8 - Let q be a natural number greater than 1 with the...Ch. 1.8 - Prob. 11ECh. 1.8 - Prob. 12ECh. 1.8 - Let a and b be nonzero integers that are...Ch. 1.8 - Let a and b be nonzero integers and d=gcd(a,b) ....Ch. 1.8 - Let a and b be nonzero integers and c be an...Ch. 1.8 - Prob. 16ECh. 1.8 - Prob. 17ECh. 1.8 - Let a and b be integers, and let m=lcm(a,b) . Use...Ch. 1.8 - The greatest common divisor of positive integers a...Ch. 1.8 - Prob. 20ECh. 1.8 - Prob. 21E
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