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Define the equivalence relation $R$ as follows: For $x,y\in\mathbb R$, $x$ is equivalent to $y$ if and only if $xy\geq 0$. Determine all of the equivalence classes of this equivalence relation.

user112464
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2 Answers2

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The title doesn't match the question. I'll use the wording from the question. As written in your question (not the title), $R$ is not an equivalence relation; $0$ is equivalent to $1$, and $-1$ is equivalent to $0$, but $-1$ is not equivalent to $1$, so $R$ is not transitive.

If you mean what the title says ($x$ is equivalent to $y$ if $xy > 0$), then $R$ is not an equivalence relation, but for a different reason. I'll leave it up to you to figure out why.

Stefan Smith
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Hint : $xy \ge 0$ is $x$ and $y$ both have same sign or one among them is zero.

GA316
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  • I have determined this is an equivalence relation on R, but am unsure of how to describe all the equivalence classes of this equivalence relation. They must be real numbers with the same sign, but is that all I need to know establish all equivalence classes? {x>0, y>0:x,y exist in R} {x<0,y<0} {x<0,y=0} {x>0,y=0} {x=0,y<0} {x=0,y>0} ? – user112464 Nov 30 '13 at 15:32
  • note that your equivalence class is a partition of $\mathbb{R}$ – GA316 Nov 30 '13 at 15:40