# Notation: is it correct to write $x \succsim y \succsim z$ before using transitivity?

Let $$\succsim$$ complete and $$x, y, z \in X$$. Suppose $$x \succsim y$$ and $$y \succsim z$$. Is it OK to write $$x \succsim y \succsim z$$ just knowing this?

Or, on the contrary, such expression automatically implies $$x \succsim z$$?

• This seems more like a math question. (Math notation specifically, so very little microeconomics is involved.) Also, it boils down to whether writing $$5 > 2 < 3$$ is okay. These are two different inequalities, but I was lazy and did not write the symbol 2 again. But this clearly does not imply anything about the relationship of 5 and 3. Oct 1, 2019 at 17:37
• Strictly speaking, $\succsim$ symbolizes a binary relation, and so should be used as one. Writing "$x\succsim y\succsim z$" as a shorthand for "$x\succsim y$ and $y\succsim z$" is an abuse of notation and should only be used within an appropriate context where confusion/misinterpretation is unlikely. Oct 1, 2019 at 21:11
• To be clear, you can't conclude that $x \succsim z$ without assuming transitivity.
– Art
Oct 2, 2019 at 16:45