# Omega

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 Nov7 asked Finding the first, last, minimal and maximal elements in these relations. Nov6 accepted How can I determine that a relation lacks a property without using a counterexample? Nov6 comment Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? If a relation used single elements, would a metric line be a good analogy...? Nov6 comment Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? Daaaaamn. Back to scratch. I really like the graph idea, I wonder if I could use that more often for the other relations I tried. Nov6 accepted Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? Nov6 asked Is my transitivity proof correct for the relation over $\mathbb{Z} \times \mathbb{Z}$ where $(a,b)R(c,d) \iff (a \le c \lor b \le d)$? Nov6 asked How can I determine that a relation lacks a property without using a counterexample? Nov6 accepted Properties of relation $R$ on $\mathbb{N} \times \mathbb{N}:\;(a,b)R(c,d) \iff a -c = b -d$ Nov5 asked Properties of relation $R$ on $\mathbb{N} \times \mathbb{N}:\;(a,b)R(c,d) \iff a -c = b -d$ Nov5 accepted Is this equation to prove that $aRb \iff a^2 - b^2 = 1$ is antisymmetric correct? Nov5 comment Is this equation to prove that $aRb \iff a^2 - b^2 = 1$ is antisymmetric correct? So it isn't antisymmetric? Or rather, it is antisymmetric since there is no counterexample? Nov5 asked Is this equation to prove that $aRb \iff a^2 - b^2 = 1$ is antisymmetric correct? Nov5 comment Proving that the relation $a \le b \iff b - a \ge 0$ is antisymmetric and total. @user01123581321345589144...: Sounds likely. It seemed like my only option at the time, though. Since both expressions are $\ge 0$, adding them should still be $\ge 0$. That was probably unnecessary I guess. Nov5 comment Proving that the relation $a \le b \iff b - a \ge 0$ is antisymmetric and total. I fear that the exercise makes no remarks about $\ge$. Nov5 asked Proving that the relation $a \le b \iff b - a \ge 0$ is antisymmetric and total. Nov5 accepted Proving a relation's inverse's properties by knowing the original's. Nov5 comment Proving a relation's inverse's properties by knowing the original's. Yeah, it seems your answer is now correct. Thank you. Nov5 comment Proving a relation's inverse's properties by knowing the original's. I'm not sure what do you mean by definitions (lol). I just quoted the exercise :(.. But thanks anyway, the insight was very helpful! Nov5 comment Proving a relation's inverse's properties by knowing the original's. For the second question, we proved in the first question that $R^{-1}$ is symmetric, so $(a,c) \in R^{-1}$, therefore the intersection $R \cap R^{-1}$ also contains $(a,c)$ since $(a,c) \in R$ as well, right? However, I'm not sure if the previous question should be treated as a separate scenario, or does it not matter? Nov4 asked Proving a relation's inverse's properties by knowing the original's.