Exercise $3\mathrm{.}3\mathrm{.}1$ : Verify the theorem from Section $3\mathrm{.}3\mathrm{.}3,$ which relates the Jac$-$ card similarity to the probability of minhashing to equal values, for the partic$-$ ular case of Fig. $3\mathrm{.}2.($a$)$ Compute the Jaccard similarity of each of the pairs of columns in Fig. $3\mathrm{.}2.!($b$)$ Compute, for each pair of columns of that figure, the fraction of the $120$ permutations of the rows that make the two columns hash to the same value. Exercise $3\mathrm{.}3\mathrm{.}1$: Verify the theorem from Section $3\mathrm{.}3\mathrm{.}3,$ which relates the Jac$-$ card similarity to the probability of minhashing to equal values, for the partic$-$ ular case of Fig. $3\mathrm{.}2.($a$)$ Compute the Jaccard similarity of each of the pairs of columns in Fig. $3\mathrm{.}2.!($b$)$ Compute, for each pair of columns of that figure, the fraction of the $120$ permutations of the rows that make the two columns hash to the same value. Si S$2$ S$4$ Element b e a d O O $11$ O S$311$ o $1$ O O o $11113\mathrm{.}3$: A permutation of the rows of Fig. $3\mathrm{.}2$