Set Theory - Equivalence Relation

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• Equivalence Classes of an Equivalence Relation

  57935 Equivalence Classes of an Equivalence Relation Modern Algebra Set Theory (II)

• Set Theory and Counting

  505185 Set Theory and Counting 1. List the ordered pairs in the equivalence relations produced by these partitions of {0,1,2,3,4,5} a) {0}, {1,2}, {3,4,5} b) {0,1}, {2,3}, {4,5} c) {0,1,2}, {3,4,5} d) {0}, {1}, {2}, {3}, {4}, {5} 2.

• Equivalence Classes

  Because, in set theory, and means intersection. Therefore, the number of equivalence classes of p'' will be equal to the intersection of n and n' classes.

• Equivalence Relation vs. Equivalence Class

  Question 1: What is an equivalence relation? Solution: A given binary relation ~ on a set A is said to be an equivalence relation if and only if it is reflexive, symmetric and transitive. Equivalently, for all a, b and c in A: • a ~ a.

• Let I be the set of all integers and let m be a fixed positive integer. Two integers a and b are said to be congruent modulo m-symbolized by a is congruent to b (mod m) - if a - b is exactly divisible by m, i.e., if a - b is an integral multiple of m. Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets.

  Show that this is an equivalence relation , describe the equivalence set, and state the number of distinct equivalence sets.

• Equivalence relations, surjective maps, partitions and fibers.

  123186 Equivalence relations, surjective maps, partitions and fibers. Consider any surjective map f from a set X onto another set Y. We can define a relation on X by x_1 ~ x_2 if f(x_1) = f(x_2). Check that this is an equivalence relation.

• Equivalence Relation and Equivalence Sets

  147741 Equivalence Relation and Equivalence Sets Topology Sets and Functions (XLVI) Functions In the set R of all real numbers, let x ~ y means that x - y is an integer.

• Equivalence Relations

  More generally: for every function f on a set X with values in a set Y endowed with an equivalence relation ~, H={(a,b) in X x X|f(a)~f(b)} is an equivalence relation. Equivalence relations are investigated.

• Equivalence relations and the empty set

  an equivalence relation? Explain. A binary relation on N is a set of pairs (n, m) in N x N. The empty set has NO elements that are NOT of the form (n, m) in N x N, so this condition is satisfied (vacuously). Therefore, it IS a binary relation.