Solutions To Mathematics Textbooks/Proofs and Fundamentals/Chapter 5

Section 5.1 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 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: .. Let , for all . Then . . . Let , for all . Then . . . Let , for all . Then . . . 5.1.2 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 . Let be the relation defined by ... Because . The geometric description of the relation class are the graph of . Let be the relation defined by ... 5.1.3 Let . Each of the following subsets of defines a relation on . Is each relation reflexive, symmetric and/or transitive? Recall: Reflexive , Symmetric , Transitive , . is symmetric only; since (1,2) and (2,1) are present. . is reflexive only; since (1,1) is present. . is reflexive only; since (1,1) is present. . is reflexive only; since (1,1) is present. . is reflexive, symmetric, and transitive; (1,1) is present showing reflexive, (1,2), (2,1) are present showing symmetric, (3,1) and (1,3) are present showing symmetric. . is none. . is reflexive only; since (1,1) is present. 5.1.4 5.1.5 5.1.6 5.1.7 5.1.8 5.1.9 5.1.10 5.1.11