Explanation As the defining the equivalence classes, the following classes is named as reflexive and transitive. Ifi,l∈ Z and j, l ∈N, is so we know that every i l , , j k ∈ Z . The relation is shown for reflexive and transitive are for every i l , j k ∈ Z have i l = i l a n d j k = j k and For every i l , j k ∈ Z have that if i l = j k a n d j k = C , then i l = C respectively. Thus, this equivalence relation unites all these pairs of positive integers with the same ratio (in order) in the same equivalence class. Each of those equivalence classes can be identified with a positive rational number and there is a bijection from the equivalence classes to the set of positive rational numbers. Define addition, subtraction, multiplication and division for the classes, inspired by the corresponding operation on rational number. For example, if the positive integer is considered then the class can be defined as I f ( m , n ) ∈ [ ( 1 , 2 ) ] R ⇔ n = 2 m . So the equivalence class is for positive inetger [ ( 1 , 2 ) ] R = { ( m , 2 m ) : m ∈ Z + } I f ( m , n ) ∈ [ ( a , b ) ] R ⇔ m / n = a / b So the equivalence class is for positive rational number ( k , a ) R ( l , b ) ⇔ k b = l a Reflexive k a = k a ⇒ ( k , a ) R ( k , a ) Symmetric : ( k , a ) R ( l , b ) ⇒ k b = l a ⇒ l a = k b ⇒ ( l , b ) R ( k , a ) Transitive : ( k , a ) R ( l , b ) ∧<

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Ch. 9.1 - t the ordered pairs in the...Ch. 9.1 - a) List all the ordered pairs in the relation R =...Ch. 9.1 - each of these relations on the set {1, 2, 3, 4},...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - ermine whether the relationRon the set of all Web...Ch. 9.1 - ermine whether the relationRon the set of all real...Ch. 9.1 - ermine whether the relationRon the set of all...Ch. 9.1 - w that the relationR=Oon a nonempty set S is...Ch. 9.1 - Show that the relationR=on the empty setS=is...Ch. 9.1 - e an example of a relation on a set that is a)...

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