Matroid, Unique Circuit Property

Question

For the uniqueness of circuit in matroid, refers to this note: http://math.mit.edu/~goemans/18433S13/matroid-notes.pdf. In the proof of Theorem 4.1, the last 2 sentences " Since S is also independent, we must have that |X| = |S| and since e ∈ C1 − f, we must have that X = S + e − f ∈ I. But this means that C2 ⊆ S + e − f = X which is a contradiction since C2 is dependent.". Can someone please explain why "|S| = |X|" and why "e ∈ C1 − f, we must have that X = S + e − f ∈ I."? I have had no clue on how it is from for hours..

Answer 1

The author states without proof just below the definition of the axioms on the first page that maximal independent sets all have the same number of members. By I2, if you had two maximal independent sets of different sizes, you could take one of the elements from the larger one and use it to increase the smaller one, which is a contradiction. S and X are both maximal independent sets of S+e so |S| = |X|

X is independent because it is created by taking an independent set C1-f and making it maximally independent - so still independent. f is not an element of X because that would recreate C1 inside it, which we know is dependent. But there are only a total of |S|+1 elements to play with so if |X|=|S| and X does not contain f, it most contain e.