## Enter Set A and Set B:

<-- Enter Set A
<-- Enter Set B

Given the following sets
A = {1,2,3,4}
B = {2,4,6,8}
find the following items:

A U B, A ∩ B, A - B
B - A, A Δ B (symmetric difference)
concatenation of A and B, |A|, |B|
Jaccard Index and Distance, and Dice's Coefficient.

##### Determine A U B:

A U B is all elements of A and B

A U B = {1,2,3,4,6,8}

##### Determine A ∩ B:

A ∩ B is all elements in A and B

A = {1,2,3,4}
B = {2,4,6,8}

A ∩ B = {2,4}

##### Determine (A - B):

All elements of A excluding A ∩ B

A = {1,2,3,4}

A - B = {1,3}

##### Determine (B - A):

All elements of B excluding A ∩ B

B = {2,4,6,8}

B - A = {6,8}

##### Determine A Δ B:

A not in B, and B not in A

A Δ B = (A - B) U (B - A)

A Δ B = {1,3} U {6,8}

A Δ B = {1,3,6,8}

##### Determine A · B:

A · B = 1,2,3,4,2,4,6,8

##### Determine cardinality of A, |A|

|A| = Number of set items

|A| = 4

##### Determine cardinality of B, |B|

|B| = Number of set items

|B| = 4

##### Determine Cartesian Product:

(x,y) where
x ∈ A and y ∈ B

{(1,2),(1,4),(1,6),(1,8),(2,2),(2,4),(2,6),(2,8),(3,2),(3,4),(3,6),(3,8),(4,2),(4,4),(4,6),(4,8)}

##### Calculate Jaccard Index J(A,B):

 J(A,B)  = |A ∩ B| |A U B|

 J(A,B)  = |2| |6|

J(A,B) = 0.33333333333333

##### Calculate Jaccard Distance J(A,B):

Jσ(A,B) = 1 - J(A,B)

Jσ(A,B) = 1 - 0.33333333333333

Jσ(A,B) = 0.66666666666667

##### Calculate Dice's Coefficient:

 s  = |A ∩ B| |A| + |B|

 s  = |2| |4| + |4|

 s  = 2 8

s = 0.25

##### Determine if A ⊂ B:

All elements of A ∈ B

1 ⊄ 2,4,6,8

2 ⊂ 2,4,6,8

3 ⊄ 2,4,6,8

4 ⊂ 2,4,6,8

Because all elements of A are not in B,
then A ⊄ B

##### Determine if B ⊂ A:

All elements of B ∈ A

2 ⊂ 1,2,3,4

4 ⊂ 1,2,3,4

6 ⊄ 1,2,3,4

8 ⊄ 1,2,3,4

Because all elements of B are not in A,
then B ⊄ A

A U B = {1,2,3,4,6,8}
A ∩ B = {2,4}
A - B = {1,3}
B - A = {6,8}
A Δ B = {1,3,6,8}
A · B = 1,2,3,4,2,4,6,8
|A| = 4
|B| = 4
Cartesian Product = {(1,2),(1,4),(1,6),(1,8),(2,2),(2,4),(2,6),(2,8),(3,2),(3,4),(3,6),(3,8),(4,2),(4,4),(4,6),(4,8)}
J(A,B) = 0.33333333333333
Jσ(A,B) = 0.66666666666667
s = 0.25
Because all elements of A are not in B,
then A ⊄ B
Because all elements of B are not in A,
then B ⊄ A

A U B = {1,2,3,4,6,8}
A ∩ B = {2,4}
A - B = {1,3}
B - A = {6,8}
A Δ B = {1,3,6,8}
A · B = 1,2,3,4,2,4,6,8
|A| = 4
|B| = 4
Cartesian Product = {(1,2),(1,4),(1,6),(1,8),(2,2),(2,4),(2,6),(2,8),(3,2),(3,4),(3,6),(3,8),(4,2),(4,4),(4,6),(4,8)}
J(A,B) = 0.33333333333333
Jσ(A,B) = 0.66666666666667
s = 0.25
Because all elements of A are not in B,
then A ⊄ B
Because all elements of B are not in A,
then B ⊄ A
##### How does the Set Notation Calculator work?
Free Set Notation Calculator - Given two number sets A and B, this determines the following:
* Union of A and B, denoted A U B
* Intersection of A and B, denoted A ∩ B
* Elements in A not in B, denoted A - B
* Elements in B not in A, denoted B - A
* Symmetric Difference A Δ B
* The Concatenation A · B
* The Cartesian Product A x B
* Cardinality of A = |A|
* Cardinality of B = |B|
* Jaccard Index J(A,B)
* Jaccard Distance Jσ(A,B)
* Dice‘s Coefficient
* If A is a subset of B
* If B is a subset of A
This calculator has 2 inputs.

### What 7 formulas are used for the Set Notation Calculator?

A Δ B = (A - B) U (B - A)
AC = U - A
A ∩ B = A + B - A U B
J(A,B) = |A ∩ B|/|A U B|
Jσ(A,B) = 1 - J(A,B)

For more math formulas, check out our Formula Dossier

### What 11 concepts are covered in the Set Notation Calculator?

cardinality
a measure of the number of elements of the set
coefficient
a numerical or constant quantity placed before and multiplying the variable in an algebraic expression
difference
the result of one of the important mathematical operations, which is obtained by subtracting two numbers
element
an element (or member) of a set is any one of the distinct objects that belong to that set. In chemistry, any substance that cannot be decomposed into simpler substances by ordinary chemical processes.
index
an indicator, sign, or measure of something
intersection
the set containing all elements of A that also belong to B or equivalently, all elements of B that also belong to A.
A ∩ B
product
The answer when two or more values are multiplied together
set
a collection of different things; a set contains elements or members, which can be mathematical objects of any kind
set notation
Ways of writing sets and their elements and properties
subset
A is a subset of B if all elements of the set A are elements of the set B
union
Combine the elements of two or more sets