efine the relation ~ on the set N of positive integers by a~b if and only if a = b(10^k) for some integer k. Prove that ~ is an equivalence relation on N.
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A: Set theory
Define the relation ~ on the set N of positive integers by a~b if and only if
a = b(10^k) for some integer k. Prove that ~ is an equivalence relation on N.
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- 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 ].29. 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 .5. Let be the relation “congruence modulo ” defined on as follows: is congruent to modulo if and only if is a multiple of , we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .
- 4. Let be the relation “congruence modulo 5” defined on as follows: is congruent to modulo if and only if is a multiple of , and we write . a. Prove that “congruence modulo ” is an equivalence relation. b. List five members of each of the equivalence classes and .Let R be the relation on the set of all people who have visited a particular Web page such that x Ry if and only if person x and person y have followed the same set of links starting at this Web page (going from Web page to Web page until they stop using theWeb). Show that R is an equivalence relation.What are the equivalence classes of the equivalence relations on {0, 1, 2, 3}?
- Let R be the relation consisting of all pairs (x, y) such that x and y are strings of uppercaseand lowercase English letters with the property that for every positive integer n, the nth charactersin x and y are the same letter, either uppercase or lowercase. Show that R is an equivalence relation.Which of these ordered pairs belongs to the "does not divide" relation on the set of positive integers, where (a, b) belongs to this relation if a and b are positive integers such that a does not divide b? (Select all that apply.)Determine whether the relation R on the set of all integers defined by the rule (x,y) Î R if and only if x ≥ y2 is reflexive, symmetric, and/or transitive? Give supporting evidence.