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Let $R_1 = \{(a,b): a \leq b^2 and \ a,b \in \Bbb R \}$ and $R_2 = \{(a,b): a \leq b^3 and \ a,b \in \Bbb R \}$ be two relations on set of real numbers. Given that $R_1,R_2$ are not transitive. Please provide me counter examples. I could not construct these.

Largest Prime
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  • What's your question? Are you trying to prove that $R_1$ and $R_2$ are not transitive? – Alann Rosas Jul 26 '21 at 07:28
  • No, I'm not trying to prove because it is given that these are not transitive. – Largest Prime Jul 26 '21 at 07:31
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    Your question is still not clear for me. Do you want an example of an non-transitive relation on the real numbers? If so what do they have to do with $R_1$ and $R_2$? – Jacobiman Jul 26 '21 at 07:39
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    @Jacobiman The way I read it, they want to construct witnesses $(a,b),(b,c)\in R_i$ that proves the relations aren't transitive. So yes, I believe they want to prove non-transitiveness. – Arthur Jul 26 '21 at 07:45
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    "Given that $R_1, R_2$ are not transitive." Is not a sentence......... "Please provide me counter examples." A counter example of what. ..... It really sounds like you are asking for examples that demonstrate that $R_1$ and $R_2$ are not transitive. ... Is that what you are asking for. – fleablood Jul 26 '21 at 07:46
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    If that's the case then for instance choose $a=1$, $b=-2$ and $c=0$. Then $a R_1 b$, $b R_1 c$ yet it not true that $a R_1 c$. Apply a similar idea for $R_2$. – Jacobiman Jul 26 '21 at 07:57
  • Ohh yes, I have found an another counter example for $R_1$, Take $a =\frac{1}{3}, b =-3, c = \frac{1}{2}$ such that $a \leq b^2, b \leq c^2, a \nleq c^2$ – Largest Prime Jul 26 '21 at 08:50

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