Exercises 52 and 53,Rnrefers to the family of equivalence relations defined inExample 5.Recall thats Rnt,wheresandtare two strings if
53. Show that the partition of the set of all identifiers in C formed by the equivalence classes of identifiers with respect to the equivalence relationR31is a refinement of the partition formed by equivalence classes of identifiers with respect to the equivalence relationR8.(Compilers for “old” C consider identifiers the same when their names agree in their first eight characters, while compilers in standard C consider identifiers the same when their names agree in their first 31. characters.)
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Discrete Mathematics and Its Applications ( 8th International Edition ) ISBN:9781260091991
- a. 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_forwardIn 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.arrow_forward29. Suppose , , represents a partition of the nonempty set A. Define R on A by if and only if there is a subset such that . Prove that R is an equivalence relation on A and that the equivalence classes of R are the subsets .arrow_forward
- What are the equivalence classes of the equivalence relations in exercise 3 (answer d, and e)arrow_forwardFor nonempty sets A, B, and C, |A| <= |B| and |B| <= |C|, then |A| <= |C|. I know this is true because numerical equivalence is an equivalence relation. So do I prove this by showing an injection from A to C?arrow_forwardFind the number of different partions of a set (a) with one element (b) with two elements (c) with four elements Prove the second part of the “Equivalence Relations and Partions” Theorem. That is, prove that given a partition of a set S the relation defined as follows “xRy if and only if x and y are in the same cell” is an equivalence relation on S. Determine, with justification, whether the following relations are equivalence relations. If so, describe the partition arising from the equivalence relation. (a) x ∼ y in R if x ≥ y (b) n ∼ m in Z if |n| = |m|. We saw in class that the residue classes modulo 3 in Z + are {1, 4, 7, 10, . . . } {2, 5, 8, 11, . . . } {3, 6, 9, 12, . . . } Write the residue classes modulo n in Z + for (a) n = 4 (b) n = 5 Simplify the given expression and write your answer in the form a + bi for a, b ∈ R. (a) i^3 (b) i^5 (c) i^23 (d) (5 + 6i)(7 − i) (e) (1 + i)^3 (f) |5 + 3i|arrow_forward
- Let σ := (R), where R is a binary relation symbol. Write down a σ-theory T whose models are exactly the equivalence relations with infinitely many classes. More precisely, for each σ-structure M := (M, RM), M ⊨ T if and only if RM is an equivalence relation on M with infinitely many equivalence classes. No proof needed but please explain the answer.arrow_forwardWhich of these relations on {0, 1, 2, 3} are equivalence relations? *arrow_forwardLet A = {a, b, c, d}. How many relations defined on A are re-flexive, symmetric and transitive and contain the ordered pairs (a, b), (b, c), (c, d)?Why?arrow_forward
- Elements Of Modern AlgebraAlgebraISBN:9781285463230Author:Gilbert, Linda, JimmiePublisher:Cengage Learning,