# Jaccard: intersection, union and distance

Subject: Natural Language Processing

Exam: TEE

Reader: https://deep-learning-study-room.vercel.app/nlp/#tee/formulas/f-vectors/u2-jaccard

Canonical: https://deep-learning-study-room.vercel.app/read/nlp/tee/formulas/f-vectors--u2-jaccard/

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.