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Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 5

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Section 5.1

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Remember the definitions:
Definition 5.3.1 Let and be non-empty sets, let be a relation from to , and let . The relation class of with respect to , denoted , is the set defined by .
Definition 2.2.1 Let and be integers. The number divides the number if there is some integer such that . If divides , we write , and we say that is a factor of , and that is divisible by .

5.1.1

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  1. Let , for all . Then
    , but by the definition of the relation is and the only elements that satisfy this property are and , since and therefore . Analogously, we have to:
    .
    .
  2. Let , for all . Then
    .
    .
    .
  3. Let , for all . Then
    .
    .
    .
  4. Let , for all . Then
    .
    .
    .

5.1.2

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  1. Let be the relation defined by .
    . Because , and therefore . The geometric description of the relation class are: the -axis.
    . Because , and therefore . The geometric description of the relation class are: the the line whose equation is .
  2. Let be the relation defined by .
    .
    . Because . The geometric description of the relation class are the graph of .
  3. Let be the relation defined by .
    .
    .

5.1.3

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Let . Each of the following subsets of defines a relation on . Is each relation reflexive, symmetric and/or transitive?

  1. . is symmetric only
  2. . is reflexive only

5.1.4

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5.1.5

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5.1.6

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5.1.7

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5.1.8

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5.1.9

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5.1.10

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5.1.11

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