Miscellaneous Set Concepts Slides to accompany Sections 1.(8 & 9) - - PowerPoint PPT Presentation

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Miscellaneous Set Concepts Slides to accompany Sections 1.(8 & 9) - - PowerPoint PPT Presentation

Dec 14, 2023 217 likes •628 views

Miscellaneous Set Concepts Slides to accompany Sections 1.(8 & 9) of Discrete Mathematics and Functional Programming Thomas VanDrunen Tree example T SugarMaple SilverMaple BoxElder BaldCypress PinOak WhitePine LiveOak BristleconePine

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SLIDE 1

Miscellaneous Set Concepts

Slides to accompany Sections 1.(8 & 9) of Discrete Mathematics and Functional Programming Thomas VanDrunen

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SLIDE 2

Tree example

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder

|I| = 6.

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SLIDE 3

Tree example

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder A C

|I| = 6.

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SLIDE 4

Tree example

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder D E

|I| = 6.

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SLIDE 5

Tree example

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder I

|I| = 6.

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SLIDE 6

Tree example

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder Q P M X

|I| = 6.

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SLIDE 7

Cardinality

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder E X I

|I| = 6. |X| = 1. |E| = 3. |I − E| = 5. |X − I| = 0.

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SLIDE 8

Disjoint

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder E M

M ∩ E = ∅. |M ∩ E| = 0.

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SLIDE 9

Not Disjoint

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder A C E D

A ∩ E = {LiveOak} = ∅. C ∩ D = {BaldCypress} = ∅.

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SLIDE 10

Pairwise disjoint

T BaldCypress BristleconePine WhitePine LiveOak PinOak SugarMaple SilverMaple BoxElder Q P M X

|I| = 6.

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SLIDE 11

Questions

(E − C) ∩ (C − E) = ({LiveOak, WhitePine, BristleconePine} −{WhitePine, BristleconePineBaldCypress}) ∩{WhitePine, BristleconePineBaldCypress} −({LiveOak, WhitePine, BristleconePine}) = {LiveOak} ∩ {BaldCypress} = ∅ Is it true that for any two sets A and B, (A − B) ∩ (B − A) = ∅?

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SLIDE 12

Questions

|M ∪ Q| = |{SugarMaple, SilverMaple, BoxElder} ∪{PinOak, LiveOak}| = |{SugarMaple, SilverMaple, BoxElder, PinOak, LiveOak}| = 5 = 3 + 2 = |{SugarMaple, SilverMaple, BoxElder}| +|{PinOak, LiveOak}| = |M| + |Q| Is it true that for any two sets A and B, |A ∪ B| = |A| + |B|?

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SLIDE 13

Questions

|C − X| = |{WhitePine, BristleconePine, BaldCypress} −{BaldCypress}| = |{WhitePine, BristleconePine}| = 2 = 3 − 1 = |{WhitePine, BristleconePine, BaldCypress}| −|{BaldCypress}| = |C| − |X| Is it true that for any two sets A and B, |A − B| = |A| − |B|?

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SLIDE 14

Cartesian Plane

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SLIDE 15

Cartesian Plane

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SLIDE 16

Cartesian Plane

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SLIDE 17

Cartesian Plane

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SLIDE 18

Cartesian Plane

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SLIDE 19

Cartesian Plane

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SLIDE 20

Cartesian Plane

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SLIDE 21

Cartesian Plane

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SLIDE 22

Cartesian Plane

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SLIDE 23

Cartesian Plane

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SLIDE 24

Cartesian Plane

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SLIDE 25

Cartesian Plane

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SLIDE 26

Cartesian Plane

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SLIDE 27

Cartesian Plane

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SLIDE 28

Cartesian Plane

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SLIDE 29

Cartesian Plane

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SLIDE 30

Cartesian Plane

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SLIDE 31

Cartesian Plane

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SLIDE 32

Cartesian Plane

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SLIDE 33

Cartesian Plane

(8.6, 5.5)

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SLIDE 34

Cartesian product

Real (Cartesian) plane R × R = {(x, y) | x, y ∈ R} Cartesian product of sets X and Y : X × Y = {(x, y) | x ∈ X and y ∈ Y } (The set of ordered pairs drawn from X and Y .)

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SLIDE 35

Cartesian product example

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SLIDE 36

Cartesian product example

{

  • ,
  • ,
  • } × {
  • ,

, ,

  • , } × {

, } (

  • ,
  • ,

)

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SLIDE 37

Cartesian product example

{

  • ,
  • ,
  • } × {
  • ,

, ,

  • , } × {

, } (

  • ,

, )

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SLIDE 38

Cartesian product example

{

  • ,
  • ,
  • } × {
  • ,

, ,

  • , } × {

, } (

  • ,
  • ,

)

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SLIDE 39

Miscellaneous Set Concepts

Slides to accompany Sections 1.(8 & 9) of Discrete Mathematics and Functional Programming Thomas VanDrunen