Let A = {1,2,3,4}, and let R be a binary relation on A×A given by: ((a, b), (c, d)) ∈ R if and only if a divides c and b divides d. Show that R is an order and draw its Hasse diagram
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Let A = {1,2,3,4}, and let R be a binary relation on A×A given by: ((a, b), (c, d)) ∈ R if and only if a divides c and b divides d. Show that R is an order and draw its Hasse diagram
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- Let be a relation defined on the set of all integers by if and only if sum of and is odd. Decide whether or not is an equivalence relation. Justify your decision.True or False Label each of the following statements as either true or false. Let be an equivalence relation on a nonempty setand let and be in. If, then.12. (See Exercise 10 and 11.) If each is identified with in prove that . (This means that the order relation defined in Exercise 10 coincides in with the original order relation in . We say that the ordering in is an extension of the ordering in .) 11. (See Exercise 10.) According to Definition 5.29, is defined in by if and only if . Show that if and only if . 10. An ordered field is an ordered integral domain that is also a field. In the quotient field of an ordered integral domain define by . Prove that is a set of positive elements for and hence, that is an ordered field. Definition 5.29 Greater than Let be an ordered integral domain with as the set of positive elements. The relation greater than, denoted by is defined on elements and of by if and only if . The symbol is read “greater than.” Similarly, is read “less than.” We define if and only if. As direct consequences of the definition, we have if and only if and if and only if . The three properties of in definition 5.28 translate at once into the following properties of in . If and then . If and then . For each one and only one of the following statements is true: . The other basic properties of are stated in the next theorem. We prove the first two and leave the proofs of the others as exercises.
- Label each of the following statements as either true or false. If R is an equivalence relation on a nonempty set A, then any two equivalence classes of R contain the same number of element.In Exercises 610, a relation R is defined on the set Z of all integers. In each case, prove that R is an equivalence relation. Find the distinct equivalence classes of R and list at least four members of each. xRy if and only if x+3y is a multiple of 4.23. Let be the equivalence relation on defined by if and only if there exists an element in such that .If , find , the equivalence class containing.
- True or False Label each of the following statements as either true or false. If is an equivalence relation on a nonempty set, then the distinct equivalence classes of form a partition of.In Exercises , a relation is defined on the set of all integers. In each case, prove that is an equivalence relation. Find the distinct equivalence classes of and list at least four members of each. 10. if and only if .