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# Determine whether the relation R on the set of all integers is reflexive, symmetric, antisymmetric, and/or transitive, where (x, y) ∈ R if and only if a) x ≠ y. b) xy ≥ 1. c) x = y + 1 or x = y − 1. d) x ≡ y (mod 7). e) x is a multiple of y. f ) x and y are both negative or both nonnegative. g) x = y². h) x ≥ y².

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DEFINITIONS

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmtric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{antisymmtric}$ if $(b,a)\in R$ and $(a,b) \in R$ implies $a=b$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

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