10.2.19 - ∣ x ∣ = ∣ y ∣ Because equality is...

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INFS 501 10.2.19 Here x A y if and only if xAy x ∣=∣ y . This problem seems hard only because of the formalities using the definitions. Actually, reflexivity, symmetry, and transitivity follow simply from the fact that equality between numbers is reflexive, symmetric, and transitive. Is A Reflexive? Answer : Yes, A is reflexive because x ∈ℝ , x = x , so x A x. Is A Symmetric? Answer : Yes, A is symmtetic, here is why. Suppose x,y ∈ ℝ and x,y  ∈ A, so
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Unformatted text preview: ∣ x ∣ = ∣ y ∣ . Because equality is symmetric, we conclude ∣ y ∣ = ∣ x ∣ and y,x ∈ A, by the definition of A. Is A Transitive? Answer : Yes, A is transitive. Here is why. Suppose x,y,z ∈ ℝ , x,y ∈ A and y,z ∈ A. This means ∣ x ∣=∣ y ∣ and ∣ y ∣=∣ z ∣ . Thu s, ∣ x ∣ = ∣ y ∣ = ∣ z ∣ , so ∣ x ∣ = ∣ z ∣ , making x,z ∈ A by the definition of A....
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This note was uploaded on 11/01/2010 for the course INFS 501 taught by Professor Ellis,w during the Spring '08 term at George Mason.

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