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Suppose $M=(X,I)$ be a matroid , $X=\{x_1,...,x_m\}$ and $$Y=\{x_i\mid \operatorname{rank}(\{x_1,...,x_i\}) > \operatorname{rank}(\{x_1,...,x_{i-1}\}) \}$$ then $Y$ is in I.

Could anyone help me with this problem? Thanks for any helps.

Mathgreek
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  • Is there a difference between $y$ and $Y$? If not, the statement "$y$ is in $I$ or $Y$ is an independent set" is a tautology. If yes, what is $y$? – Randy Marsh Dec 09 '20 at 21:18
  • @Randy Marsh Oh,it's my fault.they are the same. – Mathgreek Dec 09 '20 at 21:27
  • Do you know that $\operatorname{rank}A=|A|$ iff $A\in I$, and that for any $x\in X$, $\operatorname{rank}A\le\operatorname{rank}(A\cup{x})\le\operatorname{rank}A+1$? – Brian M. Scott Dec 09 '20 at 23:54
  • @scott yes but I use it when I want to prove something is a basis for a matroid, now we know that rank(A union {xi})=rank A+1, but how should I continue? – Mathgreek Dec 10 '20 at 06:29
  • Then we have Y has elements (A union {xi}) in I or {xi} in I because it's subset should be in I, is it correct? – Mathgreek Dec 10 '20 at 06:58

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