$(30$ points $+10$ Bonus$)$ Consider the following shingle$-$document matrix.

a$.$ Compute the signature matrix using $5$ different permutations.

b$.$ Compute all pairwise column similarities for both the Col$/$Col and Sig$/$Sig$.$

c$.$ Compute the minhash signature for each column if we use the following three hash

functions:

i$.,h1(x)=x+2$mod$6$

ii$.,h2(x)=2x-5$mod$6$

iii. $h3(x)=5x+2$mod$6$

iv$.h4(x)=2x+4$mod$6$

d$.[$Bonus$]$ Which of the above hash functions are true permutations?