Data:

X = (0, 1, 2, 4, 5, 3) Y = (5, 6, 7, 9, 10, 8)

Jaccard Distance:

$JD(x,y) = 1 - \frac{ |x \wedge y |}{|x u y|}$

$ = 1 - \frac{1}{12}$

Jacard Distance = 11/12

Cosine distance:

Cosine $ (x, y) = \frac{ x . y}{ || \times || . || \times ||}$

x . y = 0 x 5 + 1 x 6 + 2 x 7 + 4 x 9 + 5 x 10 + 3 x 8

= 0 + 6 + 14 + 36 + 50 + 24

= 130

$11 \times 11 = \sqrt{0^2 + 1^2 + 2^2 + 4^2 + 5^2 + 3^2}$

$= \sqrt{0 + 1+ 4 + 16 + 25 + 9}$

$= \sqrt{55}$

$11 y 11 = \sqrt{5^2 + 6^2 + 7^2 + 9^2 + 10^2 + 8^2}$

$= \sqrt{355}$

Cosine Distance = $5 \sqrt{781}$