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4/1/2015
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Sets and Venn DiagramsPS 9A
Complements of sets, Venn Diagrams, Venn Diagram regions, numbers in regions
Universal Sets and complements•Suppose we are only interested in the natural numbers from 1 to 20, and we want to consider subsets of this set. We say the set
• 𝑈 = 𝑥 1 ≤ 𝑥 ≤ 20, 𝑥 ∈ ℕ} is the universal set in this situation.
•U is the symbol used to represent the universal set.
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Universal Sets and complements•The complement of 𝐴, denoted 𝐴’, is the set of all elements of U which are not in A.
•Sometimes said to be the opposite of A.
•The relationship between A and A’ is:
•𝐴 ∩ 𝐴′ = ∅ why is that?
•𝐴 ∪ 𝐴′ = 𝑈 why is that?
•𝑛 𝐴 + 𝑛 𝐴′ = 𝑛(𝑈)
Universal Sets and complements•Examples:
•Find 𝐶′for the given information. Write as a set.
•A) 𝑈 = 𝑎𝑙𝑙 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠𝐶 = {𝑎𝑙𝑙 𝑜𝑑𝑑 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠}
•B) 𝐶 = 𝑥 𝑥 ≥ 2, 𝑥 ∈ ℤ , 𝑈 = ℤ
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Universal Sets and complements•You try: suppose
•𝑈 = 𝑥 −4 ≤ 𝑥 ≤ 4, 𝑥 ∈ ℤ ,
•𝐴 = 𝑥 1 ≤ 𝑥 ≤ 3, 𝑥 ∈ ℤ
•𝐵 = 𝑥 −2 ≤ 𝑥 < 1, 𝑥 ∈ ℤ
•List the elements of:
• 1) 𝐴 4) 𝐴 ∩ 𝐵
• 2) 𝐵 5) 𝐴′ ∩ 𝐵
• 3) 𝐴′ 6)𝐴′ ∪ 𝐵′
Universal Sets and complements•You try: suppose
•𝑈 = 𝑥 −4 ≤ 𝑥 ≤ 4, 𝑥 ∈ ℤ ,
•𝐴 = 𝑥 1 ≤ 𝑥 ≤ 3, 𝑥 ∈ ℤ
•𝐵 = 𝑥 −2 ≤ 𝑥 < 1, 𝑥 ∈ ℤ
Find) 𝑛(𝐵)
Find) 𝑛(𝐵 ∩ 𝐴)
Find) 𝑛 𝐴′ ∪ 𝐵 − 𝑛 𝐴′ ∩ 𝐵′
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Venn Diagrams•A Venn diagram consists of a universal set U represented by a rectangle.
•Sets within the universal set are usually represented by circles.
Venn Subsets
What is the arrow pointing to?
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What are these?
What is the arrow pointing to? Can you give me an example of a situation?
Venn Subsets
What is the arrow pointing to? Can you give me an example of a situation?
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Venn Subsets
What would you call this situation? Can you give me an example?
Venn Diagram Regions•We can use shading to show various sets on a Venn diagram.
•For example, for two intersecting sets A and B:
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Venn Diagram Regions & Numbers• There are many situations where we are only
interested in the number of elements of U that are in each region. We do not need to show all the elements on the diagram, so instead we write the number of elements in each region in brackets.
1) What is the number of the Universal set for this?
Venn Diagram Regions and Numbers
In the Venn diagram given, (3) means that there
are 3 elements in the set P ∩ Q. How many
elements are there in:
a) P b) 𝑄’c) 𝑃 ∪ 𝑄 d) 𝑃, but not 𝑄e) 𝑄, but not 𝑃 f) neither 𝑃 nor 𝑄?
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Venn Diagram Properties•𝑛 𝐴 ∩ 𝐵′ = 𝑛 𝐴 − 𝑛 𝐴 ∩ 𝐵
•𝑛 𝐴′ ∩ 𝐵 = 𝑛 𝐵 − 𝑛 𝐴 ∩ 𝐵
Problems solving with Venn Diagrams
• A squash club has 27 members. 19 have black hair, 14 have brown eyes, and 11 have both black hair and brown eyes.
1. Place this information on a Venn diagram.
2. Hence find the number of members with:1. black hair or brown eyes
2. black hair, but not brown eyes.
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Problems solving with Venn Diagrams
• A squash club has 27 members. 19 have black hair, 14 have brown eyes, and 11 have both black hair and brown eyes.
1. Place this information on a Venn diagram.
2. Hence find the number of members with:1. black hair or brown eyes
2. black hair, but not brown eyes.
What do we know?
Problems solving with Venn Diagrams
• A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 5 m. How many dive from both platforms?
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Problems solving with Venn Diagrams• A city has three football teams in the national league: A,
B, and C.
• In the last season, 20% of the city’s population saw team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play.
• Using a Venn diagram, find the percentage of the city’s population which:
1. saw only team A play
2. saw team A or team B play but not team C
3. did not see any of the teams play.
• A city has three football teams in the national league: A, B, and C.
• In the last season, 20% of the city’s population saw team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play.
• Using a Venn diagram, find the percentage of the city’s population which:
1. saw only team A play
2. saw team A or team B play but not team C
3. did not see any of the teams play.
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• A city has three football teams in the national league: A, B, and C.
• In the last season, 20% of the city’s population saw team A play, 24% saw team B, and 28% saw team C. Of these, 4% saw both A and B, 5% saw both A and C, and 6% saw both B and C. 1% saw all three teams play.
• Using a Venn diagram, find the percentage of the city’s population which:
1. saw only team A play
2. saw team A or team B play but not team C
3. did not see any of the teams play.
Homework
p.216 7B, #1 & 2
p.217 7C, #1-13odd
p.221 7D, #1,2,4-7
p.224 7F, #2-8even
p.226 7G, 1,3,4