Math 1030 Video Guide
Module 1: Logic and Numbers
Lesson 1: Sets and Venn Diagrams
Time
0:00 - 1:14
1:15 - 6:52
6:53 - 9:20
9:21 - 14:17
14:18 - 17:04
17:05 - 21:43
21:44 - 30:45
30:46 - 35:59
Segment
Description
Video 1: Sets and Venn Diagrams
Pascal, Fibonacci, and Somerville Anchor problem introduced
sets problem
Introduction to the notation and vo- Discusses notation and terms (Set,
cabulary of sets
Element, Subset, Disjoint Sets, and
Overlapping Sets) used.
Example 1: Sets
Listing elements of described sets
Introduction of Venn Diagrams and A visual way to represent the relaexample 2: Drawing Venn Diagrams tionship between sets.
Discussion of ”A Typical Venn Dia- Universal Set, Intersection, Union
gram”: More vocabulary and nota- and compliment of a set are detion of sets.
scribed and visualized with Venn Diagrams
Example 3: Using a Venn Diagram Venn Diagram is used to determine
(2 sets)
number of people fitting a given description
Example 4:Using a Venn Diagram Venn Diagram is used to determine
(three setas)
number of people fitting a given description involving three sets
Solving the Pascal, Fibonacci and Anchor problem solved.
Somerville set problem
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Connections:
The Fibonacci Set is referenced. If students are unfamiliar with this set, either let them know it
will be discussed in more detail later, or stop the video and discuss it now (at time 1:11).
Additional examples and discussion is likely to be needed sometime with the segment on the notation and vocabulary of sets.
The segment on Venn Diagrams shows many good examples, introduces the ”Universal Set,” which
is described in more detail in the next segment. In example d) (around 12:45) she states that ”generically this is how we usually draw three circles in a Venn diagram.” Some discussion of ”generically”
and the discussion of the number of regions being 23 might be needed (see note below). The last
example (Brothers, uncles, parents) provides a nice springboard for more discussion. ”Can there be
uncles that are not brothers?” ”Is this one drawn correctly?” Stop the video here (14:10) for more
discussion.
At 17:17 it is mentioned that the number of sections is 22 when working with 2 sets. Discuss
with the students why it is 2n where n is the number of sets being considered. This will come up
again in Example 4 (at 21:48) for three sets (so 23 ).
When she moves into using Venn Diagrams (around 21:43) it would be a good place to stop and do
more examples to lead the students away from a ”common sense” approach to develop the algorithm
for (A ∪ B) = A + B − (A ∩ B). This is also a good place to discuss cardinality of a set.
In example 4, using three sets, the discussion follows finding the number of people in each region. At 25:45 she quickly simplifies the arithmetic; some students may need clarification. At
26:54 she begins to find the number in the region outside all circles (but does not state where this
is until she is finished). She finishes with an overall look at how to work the problem from the
inside-out and accounting for the dual counting in overlapping region. Students will certainly need
more examples before being able to go forward independently. This is not as intuitive as with two
sets. The follow-up questions help to clarify what each region represents.
At 30:46 we return to the anchor problem. The rules are more clearly defined, as well as a discussion
on the Fibonacci sequence (from 30:58 to 31:35). Stop the video any time after the rules are given
to do the activity with the class, and then follow through with her description. The questions are
left to discuss after the video ends. The first question is meant to have the students redraw the
diagram with the number of elements in each region, rather than the actual numbers that reside
there. The second question is meant to discuss the meaning of the region that is empty.
Extensions:
A discussion on cardinality is natural when learning set notation. Directing the discussion to the
”size” of the number systems (Natural, Whole, Integers, Rational, Real, etc.) will help students
deepen their understanding of continuous versus discrete.
Why is the number of regions 2n for n overlapping sets?
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Extension is possible with other sets used in the anchor problem game. Evens, powers of other
integers, etc. could replace the ones in the problem.
Formulas and Algorithms
The main formula used is the formula for the cardinality of a union of sets: (A∪B) = A+B −(A∩B)
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Example Problems: Sets and Venn Diagrams
1 List some elements of these sets:
• First letter of the days of the week
• Integers between −π and 7.3
• Multiples of 3
2 Draw a Venn diagram that represents these pairs of sets
• Nurses and Skydivers
• Limericks and Poems
• Navy Seals and Green Berets
• Hockey players, figure skaters, women
• Brothers, uncles, parents
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3 Determine how many people are in each region of this Venn Diagram.
50 People were surveyed about a certain political ad.
30 people said they saw it on TV.
25 heard it on the radio.
12 did not see or hear the ad at all.
• How many are in the intersection of T and R?
• How many are in the union of T and R?
• How many are in the complement of R?
4 Among Benjamin Franklin’s inventions are the lightning rod, the Franklin Stove and the carriage
odometer. In an old survey of 125 people, it was found that:
56 had a Franklin Stove
70 had a lightning rod
30 had a lightning rod and carriage odometer
25 had a lightning rod and a Franklin Stove
15 had a carriage odometer and a Franklin Stove
4 had all three inventions
70 did not own a carriage odometer
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• How many people owned a carriage odometer?
• How many people owned none of his inventions?
• How many owned a lightning rod or a Franklin Stove?
• How many owned exactly two of his inventions?
5 Anchor Problem: Pascal, Fibonacci and Somerville are playing a game with their favorite
numbers.
a. Fibonacci (F) likes the number is his famous sequence: 1, 1, 2, 3, 5 ...
b. Pascal (P) like powers of 2, including 20 .
c. Somerville (S) likes perfect squares.
Rules
1. Numbers must be less than 50.
2. Play lowest number that hasn’t been played.
3. Put in proper region of Venn Diagram.
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Winner is the one with the most numbers not shared by others.
What does the diagram look like at the end of the game?
Which region has no numbers in it?
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Example Problems: Sets and Venn Diagrams Key
1 List some elements of these sets:
• First letter of the days of the week {m,t,w,f,s}
• Integers between −π and 7.3 {-3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7} or {-3, -2, -1, ..., 6, 7}
• Multiples of 3 {..., -9, -6, -3, 0, 3, 6, 9, ...}
2 Draw a Venn diagram that represents these pairs of sets
• Nurses and Skydivers
• Limericks and Poems
• Navy Seals and Green Berets
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• Hockey players, figure skaters, women
• Brothers, uncles, parents
3 Determine how many people are in each region of this Venn Diagram.
50 People were surveyed about a certain political ad.
30 people said they saw it on TV.
25 heard it on the radio.
12 did not see or hear the ad at all.
• How many are in the intersection of T and R? 17
• How many are in the union of T and R? 38
• How many are in the complement of R? 25
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4 Among Benjamin Franklin’s inventions are the lightning rod, the Franklin Stove and the carriage
odometer. In an old survey of 125 people, it was found that:
56 had a Franklin Stove
70 had a lightning rod
30 had a lightning rod and carriage odometer
25 had a lightning rod and a Franklin Stove
15 had a carriage odometer and a Franklin Stove
4 had all three inventions
70 did not own a carriage odometer
• How many people owned a carriage odometer? 55
• How many people owned none of his inventions? 10
• How many owned a lightning rod or a Franklin Stove? 101
• How many owned exactly two of his inventions? 58
5 Anchor Problem: Pascal, Fibonacci and Somerville are playing a game with their favorite
numbers.
a. Fibonacci (F) likes the number is his famous sequence: 1, 1, 2, 3, 5 ...
b. Pascal (P) like powers of 2, including 20 .
c. Somerville (S) likes perfect squares.
1. Numbers must be less than 50.
2. Play lowest number that hasn’t been played.
3. Put in proper region of Venn Diagram.
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Winner is the one with the most numbers not shared by others.
What does the diagram look like at the end of the game?
Which region has no numbers in it? F ∩ S
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