# Jaccard index

The Jaccard index, also known as the Jaccard similarity coefficient, is a statistic used for gauging the similarity and diversity of sample sets. It was developed by Paul Jaccard, originally giving the French name coefficient de communauté, and independently formulated again by T. Tanimoto. Thus, the Tanimoto index or Tanimoto coefficient are also used in some fields. However, they are identical in generally taking the ratio of Intersection over Union. The Jaccard coefficient measures similarity between finite sample sets, and is defined as the size of the intersection divided by the size of the union of the sample sets:

Jaccard index is a draft programming task. It is not yet considered ready to be promoted as a complete task, for reasons that should be found in its talk page.
 This page uses content from Wikipedia. The original article was at Jaccard index. The list of authors can be seen in the page history. As with Rosetta Code, the text of Wikipedia is available under the GNU FDL. (See links for details on variance)
J(A, B) = |A ∩ B|/|A ∪ B|

Define sets as follows, using any linear data structure:

```A = {}
B = {1, 2, 3, 4, 5}
C = {1, 3, 5, 7, 9}
D = {2, 4, 6, 8, 10}
E = {2, 3, 5, 7}
F = {8}
```

Write a program that computes the Jaccard index for every ordered pairing (to show that J(A, B) and J(B, A) are the same) of these sets, including self-pairings.

## APL

```task←{
jaccard ← (≢∩)÷(≢∪)

A ← ⍬
B ← 1 2 3 4 5
C ← 1 3 5 7 9
D ← 2 4 6 8 10
E ← 2 3 5 7
F ← ,8

'.ABCDEF' ⍪ 'ABCDEF' , ∘.jaccard⍨ A B C D E F
}
```
Output:
```. A            B            C     D     E   F
A 1 0            0            0     0     0
B 0 1            0.4285714286 0.25  0.5   0
C 0 0.4285714286 1            0     0.5   0
D 0 0.25         0            1     0.125 0.2
E 0 0.5          0.5          0.125 1     0
F 0 0            0            0.2   0     1  ```

## BQN

```Jaccard ← ≡◶⟨∊ ÷○(+´) ∊∘∾, 1⟩

a ← ⟨⟩
b ← ⟨1,2,3,4,5⟩
c ← ⟨1,3,5,7,9⟩
d ← ⟨2,4,6,8,10⟩
e ← ⟨2,3,5,7⟩
f ← ⟨8⟩

Jaccard⌜˜ ⟨a,b,c,d,e,f⟩```
Output:
```┌─
╵ 1                   0                   0     0     0   0
0                   1 0.42857142857142855  0.25   0.5   0
0 0.42857142857142855                   1     0   0.5   0
0                0.25                   0     1 0.125 0.2
0                 0.5                 0.5 0.125     1   0
0                   0                   0   0.2     0   1
┘```

## Emacs Lisp

```(let* ((v1 '(A ()
B (1 2 3 4 5)
C (1 3 5 7 9)
D (2 4 6 8 10)
E (2 3 5 7)
F (8)))
(keys1 (seq-filter (lambda (x) (not (null x)))
(cl-loop for s1 being the elements of v1
using (index idx)
collect (if (= (% idx 2) 0) s1 nil)))))

(switch-to-buffer-other-window "*similarity result*")
(erase-buffer)

(defun similarity (p1 p2)
(if (and (null p1) (null p2)) 1
(/ (float (seq-length (seq-intersection p1 p2)))
(float (seq-length (seq-uniq (seq-union p1 p2))))) ) )

(insert (format "  %s\n"
(cl-loop for s1 being the elements of keys1 concat
(format "      %s" s1))))

(cl-loop for s1 in keys1 do
(insert (format "%s %s\n" s1
(cl-loop for s2 in keys1 concat
(format "  %3.3f" (similarity (plist-get v1 s1) (plist-get v1 s2) ))))))
)
```
Output:
```        A      B      C      D      E      F
A   1.000  0.000  0.000  0.000  0.000  0.000
B   0.000  1.000  0.429  0.250  0.500  0.000
C   0.000  0.429  1.000  0.000  0.500  0.000
D   0.000  0.250  0.000  1.000  0.125  0.200
E   0.000  0.500  0.500  0.125  1.000  0.000
F   0.000  0.000  0.000  0.200  0.000  1.000
```

## Factor

Works with: Factor version 0.99 2021-06-02
```USING: assocs formatting grouping kernel math math.combinatorics
prettyprint sequences sequences.repeating sets ;

: jaccard ( seq1 seq2 -- x )
2dup [ empty? ] both? [ 2drop 1 ]
[ [ intersect ] [ union ] 2bi [ length ] bi@ / ] if ;

{ { } { 1 2 3 4 5 } { 1 3 5 7 9 } { 2 4 6 8 10 } { 2 3 5 7 } { 8 } }
[ 2 <combinations> ] [ 2 repeat 2 group append ] bi
[ 2dup jaccard "%u %u -> %u\n" printf ] assoc-each
```
Output:
```{ } { 1 2 3 4 5 } -> 0
{ } { 1 3 5 7 9 } -> 0
{ } { 2 4 6 8 10 } -> 0
{ } { 2 3 5 7 } -> 0
{ } { 8 } -> 0
{ 1 2 3 4 5 } { 1 3 5 7 9 } -> 3/7
{ 1 2 3 4 5 } { 2 4 6 8 10 } -> 1/4
{ 1 2 3 4 5 } { 2 3 5 7 } -> 1/2
{ 1 2 3 4 5 } { 8 } -> 0
{ 1 3 5 7 9 } { 2 4 6 8 10 } -> 0
{ 1 3 5 7 9 } { 2 3 5 7 } -> 1/2
{ 1 3 5 7 9 } { 8 } -> 0
{ 2 4 6 8 10 } { 2 3 5 7 } -> 1/8
{ 2 4 6 8 10 } { 8 } -> 1/5
{ 2 3 5 7 } { 8 } -> 0
{ } { } -> 1
{ 1 2 3 4 5 } { 1 2 3 4 5 } -> 1
{ 1 3 5 7 9 } { 1 3 5 7 9 } -> 1
{ 2 4 6 8 10 } { 2 4 6 8 10 } -> 1
{ 2 3 5 7 } { 2 3 5 7 } -> 1
{ 8 } { 8 } -> 1
```

```import Control.Applicative (liftA2)
import Data.List (genericLength, intersect, nub, union)
import Data.List.Split (chunksOf)
import Data.Ratio (denominator, numerator)
import Text.Tabular.AsciiArt (render)

-- The Jaccard index of two sets.  If both sets are empty we define the index to
-- be 1.
jaccard :: (Eq a, Fractional b) => [a] -> [a] -> b
jaccard [] [] = 1
jaccard xs ys = let uxs = nub xs -- unique xs
isz = genericLength \$ intersect uxs ys
usz = genericLength \$ union     uxs ys
in isz / usz

-- A table of Jaccard indexes for all pairs of sets given in the argument.
-- Associated with each set is its "name", which is only used for display
-- purposes.
jaccardTable :: Eq a => [(String, [a])] -> String
jaccardTable xs = render id id showRat
\$ Table (Group SingleLine \$ map Header names)
(Group SingleLine \$ map Header names)
\$ chunksOf (length xs)
\$ map (uncurry jaccard)
\$ allPairs sets
where names = map fst xs
sets  = map snd xs

-- Show a rational number as numerator/denominator.  If the denominator is 1
-- then just show the numerator.
showRat :: Rational -> String
showRat r = case (numerator r, denominator r) of
(n, 1) -> show n
(n, d) -> show n ++ "/" ++ show d

-- All pairs of elements from the list.  For example:
--
--   allPairs [1,2] == [(1,1),(1,2),(2,1),(2,2)]
allPairs :: [a] -> [(a,a)]
allPairs xs = liftA2 (,) xs xs

main :: IO ()
main = putStrLn \$ jaccardTable [ ("A", [] :: [Int])
, ("B", [1, 2, 3, 4,  5])
, ("C", [1, 3, 5, 7,  9])
, ("D", [2, 4, 6, 8, 10])
, ("E", [2, 3, 5, 7])
, ("F", [8])]
```
Output:
```+---++---+-----+-----+-----+-----+-----+
|   || A |   B |   C |   D |   E |   F |
+===++===+=====+=====+=====+=====+=====+
| A || 1 |   0 |   0 |   0 |   0 |   0 |
+---++---+-----+-----+-----+-----+-----+
| B || 0 |   1 | 3/7 | 1/4 | 1/2 |   0 |
+---++---+-----+-----+-----+-----+-----+
| C || 0 | 3/7 |   1 |   0 | 1/2 |   0 |
+---++---+-----+-----+-----+-----+-----+
| D || 0 | 1/4 |   0 |   1 | 1/8 | 1/5 |
+---++---+-----+-----+-----+-----+-----+
| E || 0 | 1/2 | 1/2 | 1/8 |   1 |   0 |
+---++---+-----+-----+-----+-----+-----+
| F || 0 |   0 |   0 | 1/5 |   0 |   1 |
+---++---+-----+-----+-----+-----+-----+
```

## jq

Works with: jq

Works with gojq, the Go implementation of jq In the following:

• the Jaccard index is presented as a string representing a reduced fraction, e.g. "0" or "1/7".
• sets are represented by sorted arrays with distinct elements.

Preliminaries

```def lpad(\$len): tostring | (\$len - length) as \$l | (" " * \$l)[:\$l] + .;

def gcd(a; b):
# subfunction expects [a,b] as input
# i.e. a ~ .[0] and b ~ .[1]
def rgcd: if .[1] == 0 then .[0]
else [.[1], .[0] % .[1]] | rgcd
end;
[a,b] | rgcd;```

```def rjaccardIndex(x; y):
def i(a;b): a - (a-b);
def u(a;b): a + (b - i(a;b)) | unique;

def idivide(\$i; \$j):
if \$i == 0 then "0"
else gcd(\$i;\$j) as \$d
| if \$j == \$d then "\(\$i/\$d)"
else "\(\$i/\$d)/\(\$j/\$d)"
end
end;

if (x|length) == 0 and (y|length) == "0" then "1"
else idivide( i(x;y)|length; u(x;y)|length )
end;

def a : [];
def b : [1, 2, 3, 4, 5];
def c : [1, 3, 5, 7, 9];
def d : [2, 4, 6, 8, 10];
def e : [2, 3, 5, 7];
def f : [8];

[a,b,c,d,e,f] as \$sets
| [range(0;\$sets|length) | [. + 97] | implode] as \$names
| ([""] + \$names | tidy),
(range(0; \$sets|length) as \$i
| ([\$i + 97] | implode) as \$name
| \$sets[\$i] as \$x
| \$sets | map(rjaccardIndex(\$x; .)) | tidy
| "  \(\$name): \(.)" ) ;

Output:
```        a    b    c    d    e    f
a:    0    0    0    0    0    0
b:    0    1  3/7  1/4  1/2    0
c:    0  3/7    1    0  1/2    0
d:    0  1/4    0    1  1/8  1/5
e:    0  1/2  1/2  1/8    1    0
f:    0    0    0  1/5    0    1
```

## Julia

```J(A, B) = begin i, u = length(A ∩ B), length(A ∪ B); u == 0 ? 1//1 : i // u end

A = Int[]
B = [1, 2, 3, 4, 5]
C = [1, 3, 5, 7, 9]
D = [2, 4, 6, 8, 10]
E = [2, 3, 5, 7]
F = [8]
testsets = [A, B, C, D, E, F]

println("Set A             Set B             J(A, B)\n", "-"^44)
for a in testsets, b in testsets
println(rpad(isempty(a) ? "[]" : a, 18), rpad(isempty(b) ? "[]" : b, 18),
replace(string(J(a, b)), "//" => "/"))
end
```
Output:
```Set A             Set B             J(A, B)
--------------------------------------------
[]                []                1/1
[]                [1, 2, 3, 4, 5]   0/1
[]                [1, 3, 5, 7, 9]   0/1
[]                [2, 4, 6, 8, 10]  0/1
[]                [2, 3, 5, 7]      0/1
[]                [8]               0/1
[1, 2, 3, 4, 5]   []                0/1
[1, 2, 3, 4, 5]   [1, 2, 3, 4, 5]   1/1
[1, 2, 3, 4, 5]   [1, 3, 5, 7, 9]   3/7
[1, 2, 3, 4, 5]   [2, 4, 6, 8, 10]  1/4
[1, 2, 3, 4, 5]   [2, 3, 5, 7]      1/2
[1, 2, 3, 4, 5]   [8]               0/1
[1, 3, 5, 7, 9]   []                0/1
[1, 3, 5, 7, 9]   [1, 2, 3, 4, 5]   3/7
[1, 3, 5, 7, 9]   [1, 3, 5, 7, 9]   1/1
[1, 3, 5, 7, 9]   [2, 4, 6, 8, 10]  0/1
[1, 3, 5, 7, 9]   [2, 3, 5, 7]      1/2
[1, 3, 5, 7, 9]   [8]               0/1
[2, 4, 6, 8, 10]  []                0/1
[2, 4, 6, 8, 10]  [1, 2, 3, 4, 5]   1/4
[2, 4, 6, 8, 10]  [1, 3, 5, 7, 9]   0/1
[2, 4, 6, 8, 10]  [2, 4, 6, 8, 10]  1/1
[2, 4, 6, 8, 10]  [2, 3, 5, 7]      1/8
[2, 4, 6, 8, 10]  [8]               1/5
[2, 3, 5, 7]      []                0/1
[2, 3, 5, 7]      [1, 2, 3, 4, 5]   1/2
[2, 3, 5, 7]      [1, 3, 5, 7, 9]   1/2
[2, 3, 5, 7]      [2, 4, 6, 8, 10]  1/8
[2, 3, 5, 7]      [2, 3, 5, 7]      1/1
[2, 3, 5, 7]      [8]               0/1
[8]               []                0/1
[8]               [1, 2, 3, 4, 5]   0/1
[8]               [1, 3, 5, 7, 9]   0/1
[8]               [2, 4, 6, 8, 10]  1/5
[8]               [2, 3, 5, 7]      0/1
[8]               [8]               1/1
```

## Phix

```with javascript_semantics
include sets.e

function jaccard(sequence a, b)
integer i = length(intersection(a,b)),
u = length(union(a,b))
return iff(u=0?1:i/u)
end function

constant tests = {{},               -- A
{1, 2, 3, 4, 5},  -- B
{1, 3, 5, 7, 9},  -- C
{2, 4, 6, 8, 10}, -- D
{2, 3, 5, 7},     -- E
{8}}              -- F

for i=1 to length(tests) do
for j=i to length(tests) do
string s = sprintf("J(%c,%c)",{'A'+i-1,'A'+j-1})
atom jij = jacard(tests[i],tests[j])
if i!=j then
atom jji = jacard(tests[j],tests[i])
assert(jji==jij)
s &= sprintf(" = J(%c,%c)",{'A'+j-1,'A'+i-1})
end if
printf(1,"%s = %g\n",{s,jij})
end for
end for
```
Output:
```J(A,A) = 1
J(A,B) = J(B,A) = 0
J(A,C) = J(C,A) = 0
J(A,D) = J(D,A) = 0
J(A,E) = J(E,A) = 0
J(A,F) = J(F,A) = 0
J(B,B) = 1
J(B,C) = J(C,B) = 0.428571
J(B,D) = J(D,B) = 0.25
J(B,E) = J(E,B) = 0.5
J(B,F) = J(F,B) = 0
J(C,C) = 1
J(C,D) = J(D,C) = 0
J(C,E) = J(E,C) = 0.5
J(C,F) = J(F,C) = 0
J(D,D) = 1
J(D,E) = J(E,D) = 0.125
J(D,F) = J(F,D) = 0.2
J(E,E) = 1
J(E,F) = J(F,E) = 0
J(F,F) = 1
```

## Perl

```#!/usr/bin/perl

use strict;
use warnings;

my %sets = (
A => [],
B => [1, 2, 3, 4, 5],
C => [1, 3, 5, 7, 9],
D => [2, 4, 6, 8, 10],
E => [2, 3, 5, 7],
F => [8],
);
use Data::Dump 'dd'; dd \%sets;

for my \$left (sort keys %sets )
{
for my \$right (sort keys %sets )
{
my %union;
\$union{ \$_ }++ for @{ \$sets{\$left} }, @{ \$sets{\$right} };
print "J(\$left,\$right) = ",
%union ? (grep \$_ == 2, values %union) / (keys %union) : 1, "\n";
}
}
```
Output:
```{
A => [],
B => [1 .. 5],
C => [1, 3, 5, 7, 9],
D => [2, 4, 6, 8, 10],
E => [2, 3, 5, 7],
F => [8],
}
J(A,A) = 1
J(A,B) = 0
J(A,C) = 0
J(A,D) = 0
J(A,E) = 0
J(A,F) = 0
J(B,A) = 0
J(B,B) = 1
J(B,C) = 0.428571428571429
J(B,D) = 0.25
J(B,E) = 0.5
J(B,F) = 0
J(C,A) = 0
J(C,B) = 0.428571428571429
J(C,C) = 1
J(C,D) = 0
J(C,E) = 0.5
J(C,F) = 0
J(D,A) = 0
J(D,B) = 0.25
J(D,C) = 0
J(D,D) = 1
J(D,E) = 0.125
J(D,F) = 0.2
J(E,A) = 0
J(E,B) = 0.5
J(E,C) = 0.5
J(E,D) = 0.125
J(E,E) = 1
J(E,F) = 0
J(F,A) = 0
J(F,B) = 0
J(F,C) = 0
J(F,D) = 0.2
J(F,E) = 0
J(F,F) = 1
```

## Prolog

```show([]).
show([X|Xs]):- write(X), show(Xs).

j(N,M,X):- M > 0 -> X is N/M; X is 1.

task:- L = [[], [1,2,3,4,5], [1,3,5,7,9], [2,4,6,8,10], [2,3,5,7], [8]],
forall((member(A,L), member(B,L)), (
findall(X, (member(X,A), member(X,B)), I), length(I,N),
findall(X, (member(X,B), not(member(X,A))), T), append(A,T,U), length(U,M),
j(N,M,J), show(["A = ",A,", B = ",B,", J = ",J]), nl)).
```
Output:
```?- task.
A = [], B = [], J = 1
A = [], B = [1,2,3,4,5], J = 0
A = [], B = [1,3,5,7,9], J = 0
A = [], B = [2,4,6,8,10], J = 0
A = [], B = [2,3,5,7], J = 0
A = [], B = [8], J = 0
A = [1,2,3,4,5], B = [], J = 0
A = [1,2,3,4,5], B = [1,2,3,4,5], J = 1
A = [1,2,3,4,5], B = [1,3,5,7,9], J = 0.42857142857142855
A = [1,2,3,4,5], B = [2,4,6,8,10], J = 0.25
A = [1,2,3,4,5], B = [2,3,5,7], J = 0.5
A = [1,2,3,4,5], B = [8], J = 0
A = [1,3,5,7,9], B = [], J = 0
A = [1,3,5,7,9], B = [1,2,3,4,5], J = 0.42857142857142855
A = [1,3,5,7,9], B = [1,3,5,7,9], J = 1
A = [1,3,5,7,9], B = [2,4,6,8,10], J = 0
A = [1,3,5,7,9], B = [2,3,5,7], J = 0.5
A = [1,3,5,7,9], B = [8], J = 0
A = [2,4,6,8,10], B = [], J = 0
A = [2,4,6,8,10], B = [1,2,3,4,5], J = 0.25
A = [2,4,6,8,10], B = [1,3,5,7,9], J = 0
A = [2,4,6,8,10], B = [2,4,6,8,10], J = 1
A = [2,4,6,8,10], B = [2,3,5,7], J = 0.125
A = [2,4,6,8,10], B = [8], J = 0.2
A = [2,3,5,7], B = [], J = 0
A = [2,3,5,7], B = [1,2,3,4,5], J = 0.5
A = [2,3,5,7], B = [1,3,5,7,9], J = 0.5
A = [2,3,5,7], B = [2,4,6,8,10], J = 0.125
A = [2,3,5,7], B = [2,3,5,7], J = 1
A = [2,3,5,7], B = [8], J = 0
A = [8], B = [], J = 0
A = [8], B = [1,2,3,4,5], J = 0
A = [8], B = [1,3,5,7,9], J = 0
A = [8], B = [2,4,6,8,10], J = 0.2
A = [8], B = [2,3,5,7], J = 0
A = [8], B = [8], J = 1
true.
```

## Quackery

```  [ \$ "bigrat.qky" loadfile ] now!

[ over size - space swap of
join echo\$ ]                     is recho\$       ( \$ n --> \$   )

[ dip unbuild recho\$ ]             is recho        ( x n --> \$   )

[ 0 swap witheach [ bit | ] ]      is set          (   [ --> n   )

[ & ]                              is intersection (   n --> n   )

[ | ]                              is union        (   n --> n   )

[ [] 0 rot
[ dup 0 > while
dup 1 & if
[ dip [ tuck join swap ] ]
dip 1+
1 >> again ]
2drop ]                          is items        (   n --> [   )

[ 2dup = iff [ 2drop 1 1 ] done
2dup union items size
dip [ intersection items size ]
dup 0 = if [ 2drop 0 1 ]
]                                  is jaccard      ( n n --> n/d )

[ ' [ ] set ]             constant is A            (     --> n   )
[ ' [ 1 2 3 4 5 ] set ]   constant is B            (     --> n   )
[ ' [ 1 3 5 7 9 ] set ]   constant is C            (     --> n   )
[ ' [ 2 4 6 8 10 ] set ]  constant is D            (     --> n   )
[ ' [ 2 3 5 7 ] set ]     constant is E            (     --> n   )
[ ' [ 8 ] set ]           constant is F            (     --> n   )

' [ A B C D E F ]
dup witheach
[ over witheach
[ over items 15 recho
dup items 15 recho
say "--> "
2dup jaccard
proper\$ echo\$
cr drop ]
drop
drop```
Output:
```[ ]            [ ]            --> 1
[ ]            [ 1 2 3 4 5 ]  --> 0
[ ]            [ 1 3 5 7 9 ]  --> 0
[ ]            [ 2 4 6 8 10 ] --> 0
[ ]            [ 2 3 5 7 ]    --> 0
[ ]            [ 8 ]          --> 0
[ 1 2 3 4 5 ]  [ 1 2 3 4 5 ]  --> 1
[ 1 2 3 4 5 ]  [ 1 3 5 7 9 ]  --> 3/7
[ 1 2 3 4 5 ]  [ 2 4 6 8 10 ] --> 1/4
[ 1 2 3 4 5 ]  [ 2 3 5 7 ]    --> 1/2
[ 1 2 3 4 5 ]  [ 8 ]          --> 0
[ 1 3 5 7 9 ]  [ 1 3 5 7 9 ]  --> 1
[ 1 3 5 7 9 ]  [ 2 4 6 8 10 ] --> 0
[ 1 3 5 7 9 ]  [ 2 3 5 7 ]    --> 1/2
[ 1 3 5 7 9 ]  [ 8 ]          --> 0
[ 2 4 6 8 10 ] [ 2 4 6 8 10 ] --> 1
[ 2 4 6 8 10 ] [ 2 3 5 7 ]    --> 1/8
[ 2 4 6 8 10 ] [ 8 ]          --> 1/5
[ 2 3 5 7 ]    [ 2 3 5 7 ]    --> 1
[ 2 3 5 7 ]    [ 8 ]          --> 0
[ 8 ]          [ 8 ]          --> 1
```

## Raku

```sub J(\A, \B) { A ∪ B ?? (A ∩ B) / (A ∪ B) !! A ∪ B == A ∩ B ?? 1 !! 0 }

my %p =
A => < >,
B => <1 2 3 4 5>,
C => <1 3 5 7 9>,
D => <2 4 6 8 10>,
E => <2 3 5 7>,
F => <8>,
;

.say for %p.sort;
say '';
say "J({.join: ','}) = ", J |%p{\$_} for [X] <A B C D E F> xx 2;
```
Output:
```A => ()
B => (1 2 3 4 5)
C => (1 3 5 7 9)
D => (2 4 6 8 10)
E => (2 3 5 7)
F => 8

J(A,A) = 1
J(A,B) = 0
J(A,C) = 0
J(A,D) = 0
J(A,E) = 0
J(A,F) = 0
J(B,A) = 0
J(B,B) = 1
J(B,C) = 0.428571
J(B,D) = 0.25
J(B,E) = 0.5
J(B,F) = 0
J(C,A) = 0
J(C,B) = 0.428571
J(C,C) = 1
J(C,D) = 0
J(C,E) = 0.5
J(C,F) = 0
J(D,A) = 0
J(D,B) = 0.25
J(D,C) = 0
J(D,D) = 1
J(D,E) = 0.125
J(D,F) = 0.2
J(E,A) = 0
J(E,B) = 0.5
J(E,C) = 0.5
J(E,D) = 0.125
J(E,E) = 1
J(E,F) = 0
J(F,A) = 0
J(F,B) = 0
J(F,C) = 0
J(F,D) = 0.2
J(F,E) = 0
J(F,F) = 1```

## RPL

Works with: Halcyon Calc version 4.2.7
RPL code Comment
``` ≪ → a b
≪ a 1 b SIZE FOR j
b j GET IF a OVER POS THEN DROP ELSE + END
NEXT
≫ ≫ 'UNION' STO

≪ → a b
≪ { } 1 a SIZE FOR j
a j GET IF b OVER POS THEN + ELSE DROP END
NEXT
≫ ≫ 'INTER' STO

≪ → a b
≪ a b INTER SIZE a b UNION SIZE /
≫ ≫ 'JACAR' STO
```
```UNION ( {A} {B} -- {A ∪ B} )
Scan {B}...
... and add to {A} all {B} items not already in {A}

INTER ( {A} {B} -- {A ∩ B} )
Scan {A}...
... and keep {A} items also in {B}

JACAR ( {A} {B} -- Jaccard_index )

```
Input:
```{ 1 2 3 4 5 } { 1 3 5 7 9 } JACAR
{ 1 3 5 7 9 } { 1 2 3 4 5 } JACAR
```
Output:
```2: 0.428571428571
1: 0.428571428571
```

## Wren

Library: Wren-set
Library: Wren-iterate
Library: Wren-fmt

Note that the Set object in the above module is implemented as a Map and consequently the iteration order (and the order in which elements are printed) is undefined.

```import "./set" for Set
import "./iterate" for Indexed
import "./fmt" for Fmt

var jaccardIndex = Fn.new { |a, b|
if (a.count == 0 && b.count == 0) return 1
return a.intersect(b).count / a.union(b).count
}

var a = Set.new([])
var b = Set.new([1, 2, 3, 4, 5])
var c = Set.new([1, 3, 5, 7, 9])
var d = Set.new([2, 4, 6, 8, 10])
var e = Set.new([2, 3, 5, 7])
var f = Set.new([8])
var isets = Indexed.new([a, b, c, d, e, f])
for (se in isets) {
var i = String.fromByte(se.index + 65)
var v = se.value
v = v.toList.sort() // force original sorted order
Fmt.print("\$s = \$n", i, v)
}
System.print()
for (se1 in isets) {
var i1 = String.fromByte(se1.index + 65)
var v1 = se1.value
for (se2 in isets) {
var i2 = String.fromByte(se2.index + 65)
var v2 = se2.value
Fmt.print("J(\$s, \$s) = \$h", i1, i2, jaccardIndex.call(v1, v2))
}
}
```
Output:
```A = []
B = [1, 2, 3, 4, 5]
C = [1, 3, 5, 7, 9]
D = [2, 4, 6, 8, 10]
E = [2, 3, 5, 7]
F = [8]

J(A, A) = 1
J(A, B) = 0
J(A, C) = 0
J(A, D) = 0
J(A, E) = 0
J(A, F) = 0
J(B, A) = 0
J(B, B) = 1
J(B, C) = 0.428571
J(B, D) = 0.25
J(B, E) = 0.5
J(B, F) = 0
J(C, A) = 0
J(C, B) = 0.428571
J(C, C) = 1
J(C, D) = 0
J(C, E) = 0.5
J(C, F) = 0
J(D, A) = 0
J(D, B) = 0.25
J(D, C) = 0
J(D, D) = 1
J(D, E) = 0.125
J(D, F) = 0.2
J(E, A) = 0
J(E, B) = 0.5
J(E, C) = 0.5
J(E, D) = 0.125
J(E, E) = 1
J(E, F) = 0
J(F, A) = 0
J(F, B) = 0
J(F, C) = 0
J(F, D) = 0.2
J(F, E) = 0
J(F, F) = 1
```