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.

A U B is all elements of A and B

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

A ∩ B is all elements in A __and__ B

A = {1,**2**,3,**4**}

B = {**2**,**4**,6,8}

A ∩ B = **{2,4}**

All elements of A excluding A ∩ B

A = {**1**,2,**3**,4}

A - B = **{1,3}**

All elements of B excluding A ∩ B

B = {2,4,**6**,**8**}

B - A = **{6,8}**

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}

A · B = 1,2,3,4,2,4,6,8

|A| = Number of set items

|A| = 4

|B| = Number of set items

|B| = 4

(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)}

J(A,B) = | |A ∩ B| |

|A U B| |

J(A,B) = | |2| |

|6| |

J(A,B) = **0.33333333333333**

J_{σ}(A,B) = 1 - J(A,B)

J_{σ}(A,B) = 1 - 0.33333333333333

J_{σ}(A,B) = **0.66666666666667**

s = | |A ∩ B| |

|A| + |B| |

s = | |2| |

|4| + |4| |

s = | 2 |

8 |

s = **0.25**

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**

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**

J_{σ}(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 ∩ B =

A - B =

B - A =

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) =

J

s =

Because all elements of A are not in B,

then

Because all elements of B are not in A,

then

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**

J_{σ}(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 ∩ B =

A - B =

B - A =

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) =

J

s =

Because all elements of A are not in B,

then

Because all elements of B are not in A,

then

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 J_{σ}(A,B)

* Dice‘s Coefficient

* If A is a subset of B

* If B is a subset of A

This calculator has 2 inputs.

* 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 J

* Dice‘s Coefficient

* If A is a subset of B

* If B is a subset of A

This calculator has 2 inputs.

A Δ B = (A - B) U (B - A)

A^{C} = U - A

A ∩ B = A + B - A U B

J(A,B) = |A ∩ B|/|A U B|

J_{σ}(A,B) = 1 - J(A,B)

For more math formulas, check out our Formula Dossier

A

A ∩ B = A + B - A U B

J(A,B) = |A ∩ B|/|A U B|

J

For more math formulas, check out our Formula Dossier

- 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

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