a45f3e15-f31b-47a7-b2be-b995ec3e0f25 103 Davide Carpi 4 5 false false 0 0 decimal

&[DATE] &[TIME] - &[FILENAME]

Set Operations plugin

SMath Studio 3 - appVersion 1.1.8763

INTRODUCTION

All variables and functions have a set_ prefix

A dedicated toolbox is available on the right hand side of the canvas

SETS

roster set

a list of elements

A 123422 mat :

B 5223 - 31 mat :

C 3011 - 14 mat :

D 3212 mat :

E 324114 mat :

NOTE:
- any matrix can be used as a roster set;
- duplicate entries are allowed as input; however they are counted as single items in set operations;
- anything can be an element of a roster set (numbers, strings, variables, matrices, ...)

empty set

a set without members

00 matrix 00 mat

set-builder

set definition by predicate

variables condition_1 condition_2 # condition_n set_SetBuilder

the set-builder will evaluate itself if:
- the first argument contains the membership operator and a roster set is given
- the variable given is already defined and is a roster set;
- set_Universe is defined and is a roster set;

x A set_ElementOf x 1 > x 3 ≤ set_SetBuilder 2312 mat

x y 12 mat 105 - 12 mat 10 - 512 mat 4212 mat 4 - 2 - 12 mat 41 mat set_ElementOf x 5 - > y 0 < set_SetBuilder 4 - 2 - 12 mat 105 - 12 mat 12 mat

z 5 - 5 range :

x P x abs 4 > :

z z P set_SetBuilder 5 - 512 mat

otherwise the set-builder won't evalute unless it is used in set membership/operations/subset functions

x x 1 > x 4 ≤ set_SetBuilder x x 1 > x 4 ≤ set_SetBuilder

π x x 1 > x 4 ≤ set_SetBuilder set_ElementOf 1

5 x x 1 > x 4 ≤ set_SetBuilder set_ElementOf 0

universe set

current universe

set_Universe

this set is required to be defined to evalute the set_Complement(1) function

x P x abs 4 >

QUANTIFIERS

x Q x 3 > :

for all

# # set_ForAll

z z P set_ForAll 0

x A set_ElementOf x Q set_ForAll 0

there exists

# # set_Exists

z z P set_Exists 1

x A set_ElementOf x Q set_Exists 1

does not exist

# # set_DoesNotExist

z z P set_DoesNotExist 0

x A set_ElementOf x Q set_DoesNotExist 0

there exists one and only one

# # set_Exists1

z z P set_Exists1 0

x A set_ElementOf x Q set_Exists1 1

NOTE: the enumeration of set elements works in the same way as the set-builder function

MEMBERSHIP

element of set

# # set_ElementOf

3 A set_ElementOf 1

3 B set_ElementOf 0

A 123422 mat

set contains an element

# # set_Contains

A 3 set_Contains 1

B 3 set_Contains 0

B 5223 - 31 mat

not an element of set

# # set_NotElementOf

3 A set_NotElementOf 0

3 B set_NotElementOf 1

set doesn't contains an element

# # set_DoesNotContain

A 3 set_DoesNotContain 0

B 3 set_DoesNotContain 1

D 3212 mat

E 324114 mat

SUBSETS

subset

# # set_Subset

D A set_Subset 1

A E set_Subset 1

A B set_Subset 0

superset

# # set_Superset

D A set_Superset 0

A E set_Superset 1

A B set_Superset 0

proper subset

# # set_ProperSubset

D A set_ProperSubset 1

A E set_ProperSubset 0

A B set_ProperSubset 0

proper superset

# # set_ProperSuperset

D A set_ProperSuperset 0

A E set_ProperSuperset 0

A B set_ProperSuperset 0

not subset

# # set_NotSubset

D A set_NotSubset 0

A E set_NotSubset 0

A B set_NotSubset 1

not superset

# # set_NotSuperset

D A set_NotSuperset 1

A E set_NotSuperset 0

A B set_NotSuperset 1

OPERATIONS

union

# # set_Union

# # # set_Union

A B set_Union 23 - 1234516 mat

intersection

# # set_Intersection

# # # set_Intersection

A B set_Intersection 211 mat

difference

# # set_Difference

# # # set_Difference

A B set_Difference 13413 mat

symmetric difference

# # set_SymmetricDifference

# # # set_SymmetricDifference

A B set_SymmetricDifference 23 - 134515 mat

B D set_CartesianProduct 5321 mat 5221 mat 2321 mat 2221 mat 23 - 321 mat 23 - 221 mat 16 mat

cartesian product

# # set_CartesianProduct

# # # set_CartesianProduct

power set

# set_PowerSet

D set_PowerSet 00 mat 311 mat 211 mat 3212 mat 14 mat

2 a 2 sqrt 4 sqrt 22 mat set_Cardinality 3

set cardinality

# set_Cardinality

A set_Cardinality 4

complement

# set_Complement

A set_Complement #

lastError Set 'set_Universe' is not defined.

set_Universe 53241023 mat :

A set_Complement 0512 mat

NUMBER SETS

Natural numbers

set of natural numbers

set_N

53 / set_N set_ElementOf 0

2 set_N set_ElementOf 1

2 - set_N set_ElementOf 0

π set_N set_ElementOf 0

23 i * + set_N set_ElementOf 0

5 i * - set_N set_ElementOf 0

∞ set_N set_ElementOf 0

0 set_N set_ElementOf 0

Integrers

set of integers

set_Z

53 / set_Z set_ElementOf 0

2 set_Z set_ElementOf 1

2 - set_Z set_ElementOf 1

π set_Z set_ElementOf 0

23 i * + set_Z set_ElementOf 0

5 i * - set_Z set_ElementOf 0

∞ set_Z set_ElementOf 0

0 set_Z set_ElementOf 1

Rational numbers

set of rational numbers

set_Q

53 / set_Q set_ElementOf 1

2 set_Q set_ElementOf 1

2 - set_Q set_ElementOf 1

π set_Q set_ElementOf 0

23 i * + set_Q set_ElementOf 0

5 i * - set_Q set_ElementOf 0

∞ set_Q set_ElementOf 0

0 set_Q set_ElementOf 1

Real numbers

set of real numbers

set_R

53 / set_R set_ElementOf 1

2 set_R set_ElementOf 1

2 - set_R set_ElementOf 1

π set_R set_ElementOf 1

23 i * + set_R set_ElementOf 0

5 i * - set_R set_ElementOf 0

∞ set_R set_ElementOf 0

0 set_R set_ElementOf 1

Complex numbers

set of complex numbers

set_C

53 / set_C set_ElementOf 1

2 set_C set_ElementOf 1

2 - set_C set_ElementOf 1

π set_C set_ElementOf 1

23 i * + set_C set_ElementOf 1

5 i * - set_C set_ElementOf 1

∞ set_C set_ElementOf 0

0 set_C set_ElementOf 1

Imaginary numbers

set of imaginary numbers

set_I

53 / set_I set_ElementOf 0

2 set_I set_ElementOf 0

2 - set_I set_ElementOf 0

π set_I set_ElementOf 0

23 i * + set_I set_ElementOf 0

5 i * - set_I set_ElementOf 1

∞ set_I set_ElementOf 0

0 set_I set_ElementOf 1

Whole numbers

set of whole numbers

set_W

53 / set_W set_ElementOf 0

2 set_W set_ElementOf 1

2 - set_W set_ElementOf 0

π set_W set_ElementOf 0

23 i * + set_W set_ElementOf 0

5 i * - set_W set_ElementOf 0

∞ set_W set_ElementOf 0

0 set_W set_ElementOf 1

Prime numbers

set of prime numbers

set_P

53 / set_P set_ElementOf 0

2 set_P set_ElementOf 1

2 - set_P set_ElementOf 0

π set_P set_ElementOf 0

23 i * + set_P set_ElementOf 0

5 i * - set_P set_ElementOf 0

∞ set_P set_ElementOf 0

0 set_P set_ElementOf 0

393919 set_P set_ElementOf 1

999999000001 set_P set_ElementOf 1

999999000003 set_P set_ElementOf 0

Irrational numbers

set of irrational numbers

set_J

53 / set_J set_ElementOf 0

2 set_J set_ElementOf 0

2 - set_J set_ElementOf 0

π set_J set_ElementOf 1

23 i * + set_J set_ElementOf 0

5 i * - set_J set_ElementOf 0

∞ set_J set_ElementOf 0

0 set_J set_ElementOf 0

COMBINATORICS

number of expected results: \[REGION[a30a11d0a48b4fc3b6bd84ba56071ed4]]\

n k C n! k! (n k - (! (* / :

Choose

a b c 13 mat set_Choose 00 mat a 11 mat b 11 mat c 11 mat a b 12 mat a c 12 mat b c 12 mat a b c 13 mat 18 mat

a b c 13 mat 0 set_Choose 00 mat 11 mat

30 C 1

a b c 13 mat 1 set_Choose a 11 mat b 11 mat c 11 mat 13 mat

31 C 3

a b c 13 mat 2 set_Choose a b 12 mat a c 12 mat b c 12 mat 13 mat

32 C 3

a b c 13 mat 3 set_Choose a b c 13 mat 11 mat

33 C 1

number of expected results: n!

Permute

11 range set_Permute 111 mat 11 mat

1! 1

12 range set_Permute 1212 mat 2112 mat 12 mat

2! 2

13 range set_Permute 12313 mat 21313 mat 31213 mat 13213 mat 23113 mat 32113 mat 16 mat

3! 6

P 14 range set_Permute : 123414 mat 213414 mat 312414 mat 132414 mat 14 mat

4! 24

P length 24

number of expected results:\[REGION[b74d70b454d54fce9ecef04cedf65807]]\

n! n k - (! /

13 range 2 set_Permute 1212 mat 2112 mat 1312 mat 3112 mat 2312 mat 3212 mat 16 mat

3! 32 - (! / 6

5! 53 - (! / 60

P 15 range 3 set_Permute : 12313 mat 21313 mat 31213 mat 13 mat

P length 60

TOOLS

1 - 232 0.5 * - 22 mat set_Unique 1 - 2313 mat

Remove duplicates

NOTE: a single level comparision is performed; this means that matrices like \[REGION[4d2a660a1e0d4f01a8dfba1fd52f581b]]\ and \[REGION[2f3c83728f72449498fd4d1bd33292d0]]\ are considered different items

1221 mat

2121 mat

1 - 237 - 22 mat set_Sort 7 - 1 - 2314 mat

Sort elements

1 - 237 - 22 mat set_Shuffle 1 - 7 - 2314 mat

Shuffle elements

press F9 to shuffle

1234515 mat 42 set_Shuffle 3251415 mat

deterministic shuffle

j k l m n 15 mat 42 set_Shuffle l k n j m 15 mat

A 3 - 3 range : 3 - 2 - 1 - 012371 mat

B A set_Shuffle : 0122 - 31 - 3 - 17 mat

C B set_Sort : 3 - 2 - 1 - 012317 mat

SETTINGS

set_Settings_Orientation horizontal :

alternatively: "h", "r", "row"

A B set_Union 3 - 2 - 1 - 012317 mat

set_Settings_Orientation vertical :

alternatively: "v" , "c", "column"

A B set_Union 3 - 2 - 1 - 012371 mat

set_Settings_Orientation Clear 1

A B set_Union 3 - 2 - 1 - 012317 mat