Natural Language Processing · TEE · formulas

Jaccard: intersection, union and distance

Representation & similarity

Formulas explained through full worked examples

Jaccard similarity is the fraction of shared distinct elements among all distinct elements in either set.

J(A,B)=\frac{|A\cap B|}{|A\cup B|}.

Intersection means common elements; union means all distinct elements. Repetitions do not count again. Jaccard distance is $1-J$ , not the same quantity as similarity. Two empty sets require a declared convention because the fraction becomes 0/0.

Question. Find Jaccard similarity and distance for A={cat, dog} and B={dog, fish}.

Given

Sets contain unique elements.

To find

Find Jaccard similarity and distance for A={cat, dog} and B={dog, fish}.

Formula and why it applies

$J=|A\cap B|/|A\cup B|$ ; $d=1-J$ .

Step 1. Find intersection

{dog}; count 1.

Step 2. Find union

{cat, dog, fish}; count 3.

Step 3. Divide

J=1/3≈0.333333.

Step 4. Subtract from one

Distance=1−0.333333≈0.666667.

Answer. Similarity≈0.333333; distance≈0.666667.

Question. Find Jaccard similarity and distance for A={red, blue} and B={red, blue}.

Given

Sets contain unique elements.

To find

Find Jaccard similarity and distance for A={red, blue} and B={red, blue}.

Formula and why it applies

$J=|A\cap B|/|A\cup B|$ ; $d=1-J$ .

Step 1. Find intersection

{red, blue}; count 2.

Step 2. Find union

{red, blue}; count 2.

Step 3. Divide

J=2/2≈1.000000.

Step 4. Subtract from one

Distance=1−1.000000≈0.000000.

Answer. Similarity≈1.000000; distance≈0.000000.

Question. Find Jaccard similarity and distance for A={apple} and B={orange, pear}.

Given

Sets contain unique elements.

To find

Find Jaccard similarity and distance for A={apple} and B={orange, pear}.

Formula and why it applies

$J=|A\cap B|/|A\cup B|$ ; $d=1-J$ .

Step 1. Find intersection

{}; count 0.

Step 2. Find union

{apple, orange, pear}; count 3.

Step 3. Divide

J=0/3≈0.000000.

Step 4. Subtract from one

Distance=1−0.000000≈1.000000.

Answer. Similarity≈0.000000; distance≈1.000000.

Jaccard = common unique ÷ total unique. Distance = 1 minus similarity.