Venn Diagrams and Sets


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Introduction to Venn Diagrams and Sets. Includes the Operations: Intersection, Union, Complement.

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Venn Diagrams and Sets

  1. 1. Moths Butterflies- Dull Colored - Brightly Colored- Fly at Night - Fly during Day- Rest with wings spread - Rest with wings Up- Fat Furry Bodies - Thin sleek bodies Things in Common Moths and Butterflies both have two wings and six legs, eat nectar from flowers, and originally came from a caterpillar.
  2. 2. The previous picture is called a "Venn Diagram", or "Venn-Euler Diagram".It shows the similarities and differences between Moths and Butterflies.The characteristics of moths have been placed in the green circle,and butterflies in the blue circle.Because the two insects have some common features,the two circles have been made to overlap each other.The common features are put into this common area,where the two circles cross each other and overlap.This overlapping region is called the "Intersection" of the Venn Diagram.The diagram is put inside a big rectangle which is called the "Universal Set".The "Universal Set" contains everything that we have listed about Moths and Butterflies.
  3. 3. A "Set" is a collection of items that allbelong to a well defined category or group.For example, a fish belongs to the group of animalswhich live in water.Sets are enclosed in pairs of curly brackets { }For example, we could write the following for the set ofour favorite animals which live in water:Water Animals = { Fish, Eel, Platypus }
  4. 4. Some Venn Diagrams Do Not Overlap. Water Animals = { fish, eel, platypus }Let’s also have another set of our favorite animalswritten like this: Two Legged Animals = { bird, bat }When we draw the Venn Diagram for these two sets,there is nothing in common between the two groups. (See next slide)
  5. 5. Two Legged Animals Water Animals Fish Bird eel Bat PlatypusThese two sets are “Mutually Exclusive”. An item belongs in onecircle set, or the other. There are no items that belong to both groups.
  6. 6. Two Legged Animals Water Animals Fish Bird eel Bat PlatypusThe “complement” of a set is everything not in that set. (Symbol A’) The complement of Water Animals is the set: Not Water Animals = { Bird, Bat }
  7. 7. Lets change our previous example so that it nowcontains the following Universal set of Animals:E = Everything = { Fish, Eels, Platypus, Penguins, Eagles, Bats }We are going to use a Venn diagram to dividethese animals into the following two sets:"Water Animals" and "Two Legged Animals" . (Continued Next slide)
  8. 8. When we do this, we find that Penguins belong in bothgroups:E = Everything = { Fish, Eels, Platypus, Penguins, Eagles, Bats }Water Animals = { Fish, Eels, Platypus, Penguins }Two Legged Animals = { Penguins, Eagles, Bats }This means that on our Venn Diagram, we will need to havetwo overlapping circles. We then need to put Penguins inthe middle, so that they are inside both circles. (Continued Next slide)
  9. 9. Fish Eagles Eels Penguins Bats PlatypusTwo Legged Animals Water Animals
  10. 10. The union of two sets is everything that is contained withinthe two circles all combined together. ** Each item in the Union Set is only listed once **For our Venn Diagram of Two Legged Animals and WaterAnimals, we have:{ Two Legged Animals } UNION { Water Animals } ={ Fish, Eels, Platypus, Penguins, Eagles, Bats }Union is written using a big "U" symbol, or the word "OR". (See Union Diagram on Next Slide)
  11. 11. Fish Eagles Eels Penguins Bats PlatypusTwo Legged Animals Water Animals
  12. 12. The Difference Between "OR" and "AND" is very important.The word "OR" is confusing for “Union”, because we oftenthink that our Union set should be everything in one circleAND everything in the other circle.“OR” actually means that everything in our Union answer setis either in one circle, OR in the other circle.The word "AND" is used for "Intersection“ of sets, andmeans that the item is contained in both sets.Eg. For our previous example, Penguins are in the sets:{ Two Legged Animals } AND { Water Animals }.
  13. 13. Counts of Items Diagram There are 3 animals that Fish live in water, but all 3 of Eagles these are not animals that Eels Penguins are “Two Legged” . Bats Platypus Two Legged Animals Water Animals 2 3 Counts of Items Diagram 1There are 2 animals whichare two legged , but theydo not go into water at all. Two Legged Animals Water Animals
  14. 14. Intersection can be written as:{ Two Legged Animals } Intersection { Water Animals }{ Two Legged Animals } AND { Water Animals }{ Two Legged Animals } ∩ { Water Animals } T ∩ W
  15. 15. Union can be written as:{ Two Legged Animals } Union { Water Animals }{ Two Legged Animals } OR { Water Animals }{ Two Legged Animals } U { Water Animals } T U W
  16. 16. Complement can be written as:Complement of { Two Legged Animals } or { Two Legged Animals } „ or T„ or NOT in TComplement of { Water Animals } or { Water Animals } „ or W„ or NOT in W
  17. 17. Mutually Exclusive (or “ME”) sets:Have Nothing in CommonHave No Intersection (Intersection = Null Set or Φ )We can do Union and combine items in “ME” setsWe can do the Complements of “ME” sets“ME” sets are also called “Disjoint” sets.
  18. 18. Subsets that are inside other sets:Have everything in CommonAll the items in one group, also belong to the other groupHave an Intersection that is the entire subsetAre drawn as a circle inside a circle
  19. 19. slides are exclusive Copyright of Passy’s World of Mathematics