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  3. please only do number 8 number 7 is there for...

Question: please only do number 8 number 7 is there for...

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matroid (see Exercise 7 for the definition). A matroid of rank r is called a paving matroid if all of its circuits Paving matroid have size at least r. (a) Show that if C is a circuit of any rank r matroid, then (CI S (b) Which of the following are paving matroids? Explain your answer by finding the rank of the matroid along with the sizes of all circuits. i) The Fano matroid F; see Figure 1.14. (ii) The non-Fano matroid F7; see Figure 1.15. (iii) The matroid see Figure 1.35 below. (8) This exercise continues the investigation of paving matroids - see Exercise 7 (a) Show that the uniform matroid Urn is paving for all r, n with (b) Show that all simple matroids of rank 3 are paving. 03r<n. eGive an example of a simple rank 4 matroid that is not paving (c) avin in a matroid M. i.e., no

Please only do number 8! Number 7 is there for reference.

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