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$.$ h$1($x$)=$ x $+2$ mod $6$

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

iii. h$3($x$)=5$x $+2$ mod $6$

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

Which of the above hash functions are true permutations?