Let
” defined on
a. Prove that “congruence modulo
” is an equivalence relation.
b. List five members of each of the equivalence classes
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Chapter 1 Solutions
Elements Of Modern Algebra
- Let R be the relation defined on the set of integers by aRb if and only if ab. Prove or disprove that R is an equivalence relation.arrow_forward21. A relation on a nonempty set is called irreflexive if for all. Which of the relations in Exercise 2 are irreflexive? 2. In each of the following parts, a relation is defined on the set of all integers. Determine in each case whether or not is reflexive, symmetric, or transitive. Justify your answers. a. if and only if b. if and only if c. if and only if for some in . d. if and only if e. if and only if f. if and only if g. if and only if h. if and only if i. if and only if j. if and only if. k. if and only if.arrow_forwardLet 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.arrow_forward
- 6. Prove that if is a permutation on , then is a permutation on .arrow_forwardLabel each of the following statements as either true or false. Let R be a relation on a nonempty set A that is symmetric and transitive. Since R is symmetric xRy implies yRx. Since R is transitive xRy and yRx implies xRx. Hence R is alsoreflexive and thus an equivalence relation on A.arrow_forwarda. Let R be the equivalence relation defined on Z in Example 2, and write out the elements of the equivalence class [ 3 ]. b. Let R be the equivalence relation congruence modulo 4 that is defined on Z in Example 4. For this R, list five members of equivalence class [ 7 ].arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,