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M1 for matroids

M1 for matroids

av Martin Vilcans -
Antal svar: 1

I'm confused about the rule M1 given in the slides (Part5-Slides, page 4) where it is:

(M1) F ≠ ∅ is a collection of subsets of the set R, i.e., F ⊆ P(R).

but in the notes (Part5-Script, page 9) it says:

(M1) ∅ ∈ F

which matches definitions I've found elsewhere: (I1) at https://en.wikipedia.org/wiki/Matroid and the second definition at https://mathworld.wolfram.com/Matroid.html).

∅ ∈ F implies F ≠ ∅, but it's more specific. But at the same time M2 (closed w.r.t. inclusion) imply ∅ ∈ F anyway. Whichever version of M1 you choose doesn't seem to matter because together with M2, they're equivalent. If I'm supposed to prove that something is a matroid in an exercise, can I just use whatever version of M1 that is more convenient?

Som svar till Martin Vilcans

Re: M1 for matroids

av Marc Hellmuth -
indeed, that might be confusing.

However, for independent systems (and thus for matroids) M2 always holds.
In this case F ≠ ∅ and ∅ ∈ F are equivalent.

As you pointed out above: ∅ ∈ F implies F ≠ ∅

Moreover, if F ≠ ∅ it follows that some X is in F and by (M2) all of its subsets. Since ∅ ⊆ X it follows that ∅ ∈ F .

Hence if M2 is satisfied then ∅ ∈ F if and only if F ≠ ∅