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
Final Answer
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
What is the Answer?
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?
* 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?
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)
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
