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Transcript of DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5,...
![Page 1: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/1.jpg)
DP SL StudiesChapter 7
Sets and Venn Diagrams
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DP Studies Chapter 7 Homework
Section A: 1, 2, 4, 5, 7, 9
Section B: 2, 4
Section C: 1, 2, 4, 5
Section D: 1, 4, 5, 6
Section E: 1, 4
Section F: 1, 3, 4, 7, 8
Section G: 2, 6, 8
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Contents: Sets and Venn diagrams
• A Sets• B Set builder notation• C Complements of sets• D Venn diagrams• E Venn diagram regions• F Numbers in regions• G Problem solving with Venn diagrams
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A. Set Notations
A set is a collection of numbers or objects.
Examples:
1. the set of all digits which we use to write numbers is
{0, 1, 2, 3, 4, 5, 6, 7, 8, 9}.
2. set of all vowels, then V = {a, e, i, o, u}.
![Page 5: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/5.jpg)
A. Set Notations
The numbers or objects in a set are called the elements or members of the set.
Examples:
1. So, for the set A = {1, 2, 3, 4, 5, 6, 7} we can say
4 e A (4 is an element of set A), but 9 e A
(9 is not an element of set A).
2. For the set of all vowels, V = {a, e, i, o, u}, we can
say a e V (a is an element of set V), but b e V
(b is not an element of set V).
![Page 6: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/6.jpg)
A. Set Notations
The set { } or 0 is called the empty set and contains no elements.
Example
Let A be the set of all NBA players who are 10 feet tall.
A = {}
![Page 7: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/7.jpg)
A. Set Notations
Special number sets
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A. Set Notations
The number of elements in set A is written n(A).
Example:
the set A = {2, 3, 5, 8, 13, 21} has 6 elements, so we write n(A) = 6.
![Page 9: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/9.jpg)
A. Set Notations
A set which has a finite number of elements is called a finite set.
Example:
1. A = {2, 3, 5, 8, 13, 21} is a finite set.
2. Ø is also a finite set, since n(Ø) = 0.
![Page 10: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/10.jpg)
A. Set Notations
Infinite sets are sets which have infinitely many elements.
Example:
1. the set of positive integers {1, 2, 3, 4, ....} does not have a
largest element, but rather keeps on going forever. It is
therefore an infinite set.
2. the sets N , Z , Z+, Z – , Q , and R are all infinite sets.
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A. Set Notations
Suppose P and Q are two sets. P is a subset of Q if every element
of P is also an element of Q. We write P Q.
Example:
{2, 3, 5} {1, 2, 3, 4, 5, 6} as every element in the first set is
also in the second set.
We say P is a proper subset of Q if P is a subset of Q but is
not equal to Q.
We write P Q.
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A. Set Notations
If P and Q are two sets then
P Q is the intersection of P and Q, and consists of
all elements which are in both P and Q.
P Q is the union of P and Q, and consists of all
elements which are in P or Q.
Examples:
1. If P = {1, 3, 4} and Q = {2, 3, 5} then P Q = {3} and
P Q = {1, 2, 3, 4, 5}
2. The set of integers is made up of the set of negative
integers, zero, and the set of positive integers.
Z = (Z – {0}Z +)
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A. Set Notations
Two sets are disjoint or mutually exclusive if they have no elements in common.
Example:
Set A = {0, 2, 4, 6, 8} and Set B = {1, 3, 5, 7}
Set A and Set B are disjoint or mutually exclusive
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A. Set Notations
Example 1:
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A. Set Notations
Solution to Example 1:
![Page 16: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/16.jpg)
B: Set Builder Notation
Reading a set notation:
A = {x | -2 < x < 4, x e Z}
“the set of all x such that x is an integer between -2 and 4, including -2 and 4.”
We can represent A on a number line as:
A is a finite set, and n(A) = 7.
such thatthe set of all
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B: Set Builder Notation
Reading a set notation:
B = {x | -2 < x < 4, x e R}
“the set of all real x such that x is greater than or equal to -2 and less than 4.”
We represent B on a number line as:
B is an infinite set, and n(B) = ∞
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B: Set Builder Notation
Example 2:
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Solution to example 2:
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C. Complement s of sets
The symbol U is used to represent the universal set under consideration.
Example:
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 e N } is the universal set in this
situation.
![Page 21: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/21.jpg)
C. Complement s of sets
The complement of A, denoted A’, is the set of all elements of U which are not in A.
Example:
If the universal set U = {1, 2, 3, 4, 5, 6, 7, 8}, and the
set A = {1, 3, 5, 7, 8}, then the complement of A is
A’ = {2, 4, 6}.
![Page 22: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/22.jpg)
C. Complement s of sets
Three obvious relationships are observed connecting A and A’.
1. A A’ = Ø as A’ and A have no common members.
2. A A’ = U as all elements of A and A’ combined make
up U.
3. n(A) + n(A’) = n(U)
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C. Complement s of sets
Example 3:
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C. Complement s of sets
Solution to example 3:
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C. Complement s of sets
Example 4:
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C. Complement s of sets
Solution to example 4:
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C. Complement s of sets
Example 5:
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C. Complement s of sets
Solution to example 5:
![Page 29: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/29.jpg)
D. Venn Diagrams
Venn diagrams are often used to represent sets of objects, numbers, or things.
A Venn diagram consists of a universal set U represented by a rectangle.
Sets within the universal set are usually represented by circles.
Example:
![Page 30: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/30.jpg)
D. Venn Diagrams
Example of a universe set, U = {2, 3, 5, 7, 8}, A = {2, 7, 8}, and
A’ = {3, 5}.
![Page 31: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/31.jpg)
D. Venn Diagrams
SUBSETS
If B A then every element of B is also in A. The circle representing B is placed within the circle representing A.
![Page 32: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/32.jpg)
INTERSECTION
A B consists of all elements common to both A and B.
It is the shaded region where the circles representing A and B
overlap.
![Page 33: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/33.jpg)
D. Venn Diagrams
UNION
A B consists of all elements in A or B or both. It is the shaded region which includes both circles.
![Page 34: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/34.jpg)
D. Venn Diagrams
DISJOINT OR MUTUALLY EXCLUSIVE SETS
Disjoint sets do not have common elements. They are represented by non-overlapping circles.
For example, if A = {2, 3, 8} and B = {4, 5, 9} then A B = Ø.
![Page 35: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/35.jpg)
D. Venn Diagrams
Example 6:
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Solution to example 6:
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Example 7:
![Page 38: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/38.jpg)
Solution to example 7:
![Page 39: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/39.jpg)
E. Venn Diagram Region
The shading representations of Venn Diagrams.
![Page 40: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/40.jpg)
Example 8:
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Solution to example 8:
![Page 42: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/42.jpg)
F. Numbers in Regions
The four regions of the Venn Diagram that contains two overlapping of sets A and B.
![Page 43: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/43.jpg)
F. Numbers in Regions
Example 9:
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F. Numbers in Regions
Solution to Example 9:
![Page 45: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/45.jpg)
F. Numbers in Regions
Venn diagrams allow us to easily visualize identities such as
n(A B’) = n(A) – n(A B) and
n(A’ B) = n(B) – n(A B)
![Page 46: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/46.jpg)
F. Numbers in Regions
Example 10:
Given n(U) = 30, n(A) = 14, n(B) = 17, and n(A B) = 6, find:
a. n(A B) b. n(A, but not B)
![Page 47: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/47.jpg)
F. Numbers in Regions
Solution to example 10:
![Page 48: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/48.jpg)
G. Problem solving with Venn Diagrams
Example 11:
A squash club has 27 members. 19 have black hair, 14 have
brown eyes, and 11 have both black hair and brown eyes.
a. Place this information on a Venn diagram.
b. Hence find the number of members with:
i. black hair or brown eyes
ii. black hair, but not brown eyes.
![Page 49: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/49.jpg)
G. Problem solving with Venn Diagrams
Solution to example 11:
![Page 50: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/50.jpg)
G. Problem solving with Venn Diagrams
Example 12:
A platform diving squad of 25 has 18 members who dive from 10 m and 17 who dive from 4 m. How many dive from both platforms?
![Page 51: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/51.jpg)
G. Problem solving with Venn Diagrams
Solution to example 12:
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G. Problem solving with Venn Diagrams
Example 13:
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:
a. saw only team A play
b. saw team A or team B play but not team C
c. did not see any of the teams play.
![Page 53: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/53.jpg)
Solution to example 13:
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Solution to example 13:
![Page 55: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/55.jpg)
Solution to example 13:
![Page 56: DP SL Studies Chapter 7 Sets and Venn Diagrams. DP Studies Chapter 7 Homework Section A: 1, 2, 4, 5, 7, 9 Section B: 2, 4 Section C: 1, 2, 4, 5 Section.](https://reader036.fdocuments.us/reader036/viewer/2022062421/56649dd95503460f94ace333/html5/thumbnails/56.jpg)