a) Explanation Given Information: Let S be an infinite set and let x be an element not in S . Explanation: Proof: Let s 1 , s 2 , s 3 , ........ be countably infinite subset of S . Define f : S → S ∪ { x } by f ( s 1 ) = x f ( s k + 1 ) = s k for k ≥ 1 f ( s ) = s if s ∉ { s 1 , s 2 , s 3 , .... } . f ( s 1 ) = x , are one-one functions b) To determine Toprove: The open and half-open intervals ( 0 , 1 ) and ( 0 , 1 ] have the same cardinality.

3rd Edition

ISBN: 9780131679955

Author: Edgar G. Goodaire

Publisher: Prentice Hall

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Chapter 3.3, Problem 24E

To determine

Toprove: S and S∪{x} are sets of the same cardinality

To determine

Toprove: The open and half-open intervals (0,1) and (0,1] have the same cardinality.

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Chapter 3.3, Problem 23E

Chapter 3.3, Problem 25E

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Chapter 3 Solutions

Discrete Mathematics with Graph Theory

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

Ch. 3.1 - Prob. 11TFQ Ch. 3.1 - Prob. 12TFQ Ch. 3.1 - Determine whether each of the following relation...Ch. 3.1 - 2. Suppose A is the set of students currently...Ch. 3.1 - Prob. 3E Ch. 3.1 - Prob. 4E Ch. 3.1 - Prob. 5E Ch. 3.1 - Prob. 6E Ch. 3.1 - Prob. 7E Ch. 3.1 - Prob. 8E Ch. 3.1 - Prob. 9E Ch. 3.1 - Prob. 10E Ch. 3.1 - Prob. 11E Ch. 3.1 - Prob. 12E Ch. 3.1 - Prob. 13E Ch. 3.1 - Define g:ZB by g(x)=|x|+1. Determine (with...Ch. 3.1 - Define f:AA by f(x)=3x+5. Determine (with reasons)...Ch. 3.1 - 16. Define by . Determine (with reasons) whether...Ch. 3.1 - Prob. 17E Ch. 3.1 - Prob. 18E Ch. 3.1 - Prob. 19E Ch. 3.1 - Define f:RR by f(x)=3x3+x. Graph f to determine...Ch. 3.1 - 21. (a) Define by . Graph g to determine whether g...Ch. 3.1 - Prob. 22E Ch. 3.1 - 23. Let a, b, c be real numbers and define by ....Ch. 3.1 - 24. For each of the following, find the largest...Ch. 3.1 - Prob. 25E Ch. 3.1 - Let S be a set containing the number 5. Let...Ch. 3.1 - Prob. 27E Ch. 3.1 - Prob. 28E Ch. 3.1 - Prob. 29E Ch. 3.1 - Prob. 30E Ch. 3.1 - Prob. 31E Ch. 3.1 - Prob. 32E Ch. 3.1 - Prob. 33E Ch. 3.1 - Prob. 34E Ch. 3.2 - True/False Questions The function defines by ...Ch. 3.2 - True/False Questions The function f:ZZ defines by...Ch. 3.2 - Prob. 3TFQ Ch. 3.2 - Prob. 4TFQ Ch. 3.2 - Prob. 5TFQ Ch. 3.2 - Prob. 6TFQ Ch. 3.2 - Prob. 7TFQ Ch. 3.2 - Prob. 8TFQ Ch. 3.2 - Prob. 9TFQ Ch. 3.2 - Prob. 10TFQ Ch. 3.2 - Let . Find the inverse of each of the following...Ch. 3.2 - 2. Define by . Find a formula for . Ch. 3.2 - Define f:(,0][0,) by f(x)=x2. Find a formula for...Ch. 3.2 - 4. Define by . Find a formula for . Ch. 3.2 - Prob. 5E Ch. 3.2 - Prob. 6E Ch. 3.2 - Show that each of the following functions f:AH is...Ch. 3.2 - Prob. 8E Ch. 3.2 - Prob. 9E Ch. 3.2 - Prob. 10E Ch. 3.2 - 11. Let and define functions by and . Find (a) ...Ch. 3.2 - Prob. 12E Ch. 3.2 - Prob. 13E Ch. 3.2 - Prob. 14E Ch. 3.2 - Prob. 15E Ch. 3.2 - Prob. 16E Ch. 3.2 - 17. Let A denote the set . Let i denote the...Ch. 3.2 - Prob. 18E Ch. 3.2 - Prob. 19E Ch. 3.2 - Prob. 20E Ch. 3.2 - Prob. 21E Ch. 3.2 - Prob. 22E Ch. 3.2 - Prob. 23E Ch. 3.2 - Prob. 24E Ch. 3.2 - Is the composition of two bijective functions...Ch. 3.2 - 26. Define by . (a) Find the values of . (b) Guess...Ch. 3.2 - Prob. 27E Ch. 3.2 - Prob. 28E Ch. 3.3 - True/False Questions If sets A and B are in...Ch. 3.3 - Prob. 2TFQ Ch. 3.3 - Prob. 3TFQ Ch. 3.3 - Prob. 4TFQ Ch. 3.3 - True/False Questions If A and B are finite sets...Ch. 3.3 - True/False Questions If the conditions of...Ch. 3.3 - Prob. 7TFQ Ch. 3.3 - Prob. 8TFQ Ch. 3.3 - Prob. 9TFQ Ch. 3.3 - Prob. 10TFQ Ch. 3.3 - Prob. 1E Ch. 3.3 - At first glance, the perfect squares 1, 4, 9, 16,...Ch. 3.3 - Prob. 3E Ch. 3.3 - Prob. 4E Ch. 3.3 - Prob. 5E Ch. 3.3 - Prob. 6E Ch. 3.3 - Prob. 7E Ch. 3.3 - Prob. 8E Ch. 3.3 - Prob. 9E Ch. 3.3 - Prob. 10E Ch. 3.3 - Prove that the notion of same cardinality is an...Ch. 3.3 - Prob. 12E Ch. 3.3 - Prob. 13E Ch. 3.3 - Prob. 14E Ch. 3.3 - Prob. 15E Ch. 3.3 - Prob. 16E Ch. 3.3 - Prob. 17E Ch. 3.3 - Prob. 18E Ch. 3.3 - Prob. 19E Ch. 3.3 - Prob. 20E Ch. 3.3 - Prob. 21E Ch. 3.3 - 22. Given an example of each of the following or...Ch. 3.3 - Prob. 23E Ch. 3.3 - Prob. 24E Ch. 3.3 - Prove that the points of a plane and the points of...Ch. 3.3 - Prob. 26E Ch. 3.3 - 27. (a) Show that if A and B are countable sets...Ch. 3.3 - Prob. 28E Ch. 3.3 - 29. Let S be the set of all real numbers in the...Ch. 3.3 - Let S be the set of all real numbers in the...Ch. 3.3 - Prob. 31E Ch. 3 - Define by . Determine whether f is one-to-one. Ch. 3 - Let f={(1,2),(2,3),(3,4),(4,1)} and...Ch. 3 - Prob. 3RE Ch. 3 - Prob. 4RE Ch. 3 - 5. Answer these questions for each of the given...Ch. 3 - Prob. 6RE Ch. 3 - Prob. 7RE Ch. 3 - Prob. 8RE Ch. 3 - Prob. 9RE Ch. 3 - Prob. 10RE Ch. 3 - Prob. 11RE Ch. 3 - Prob. 12RE Ch. 3 - Prob. 13RE Ch. 3 - Prob. 14RE Ch. 3 - Prob. 15RE Ch. 3 - Prob. 16RE Ch. 3 - Prob. 17RE Ch. 3 - Prob. 18RE Ch. 3 - Prob. 19RE Ch. 3 - Let S be the set of all real numbers in the...Ch. 3 - Prob. 21RE

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Toprove: S and S ∪ { x } are sets of the same cardinality

Solution Summary: The author explains that S and Scup leftxright are sets of the same cardinality.

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Toprove: S and S∪{x} are sets of the same cardinality

Toprove: The open and half-open intervals (0,1) and (0,1] have the same cardinality.

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Chapter 3 Solutions