Math 1030 Video Guide
Module 1: Logic and
Numbers
Teacher Lesson Guide
Lesson
1: L:Sets
and
Venn Diagrams
Module
Logic and
Numbers
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
Lesson 1: Sets and Venn Diagrams
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
Suggested breaks in videos
6:52 Set theory vocabulary is not very demanding, but it requires familiarity, so this is a good place to introduce a
collection of set problems as in discussion problems 1 and 2.
14:17 This introduction to Venn diagrams is pretty speedy and this is a good time (if not earlier) to ask the students
in the class to create their own collection of sets and Venn diagrams showing intersections and relative sizes.
21:43 A numerical example has just been worked out, but it seems more intuitive than systematic. Here, working
more problems (see discussion problems) can convey to the class the organization system that works.
30:45 As we move to more sets, the algebra becomes more complicated, but it still can be systematized.
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 are likely to be needed sometime with the segment on the nota- tion and
vocabulary of sets.
The segment on Venn Diagrams shows many good examples, introduces the Universal Set, and described in
greater detail in the next segment. In example d) (around 12:45) the narrator states that “gener- ically this is
how we usually draw three circles in a Venn diagram.” Some discussion of “generically” (in this case, in general
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position, meaning that the diagram should not portray any information not already given) 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 at (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 (thus 23 ). the narrator moves into examples using Venn Diagram representation (around 21:43) is 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) ,
where, for a set S , #(S ) is the number of elements in S (the cardinality of S ).
In example 4, she moves to a counting problem involving 3 sets and at. 25:45 she quickly simplifes the arithmetic.
Some students may need clarification. At 26:54 she begins to find the cardinality of the region outside all circles
(the number of people not in any of the specified sets). She finishes with an overall look at how to work the
problem from the inside-out and accounting for the dual counting in overlapping regions. Students will certainly
need more examples before being able to go forward independently. This is not as intuitive as it is with two sets.
The follow-up questions help to clarify what each region represents.
At 30:46 we return to the anchor problem. The problem is more clearly defined, as well as a discussion of 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 for discussion after the video ends.
The first question is meant to have the students redraw the diagram with the number of elements in each region.
The second question leads to a discussion 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?
This would be a good time to repeat the anchor problem with other sets and numbers.
Vocabulary, Formulas and Algorithms
Elements of Set Theory: Here, we are concerned with a research project or a study of a collection of things - all
possible things that can enter into our discourse.We cal the collection of all those things that are being considered
in our research the Universal Set, denoted it by U.
The individuals in U are called its elements. A set S is a collection of things from the universal set: it could
be everything, some collection defined by a specific property, or nothing. The expression x ∈ S says that the
individual x is a member of the set S ., and x < S asserts that x is not a member of S . The Greek letter Φ is used
to denote the set with no members.
Given two sets S and T :
S ⊂ T asserts that every member of S is also a member of T .
S ∩ T is the set of elements that are members of both S and T .
S ∪ T is the set of elements that are either in S or in T .
S − T is the set of elements that are in S , but not in T .
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We say that S and T are disjoint if there is nothing that is a member of both S and T . This is the same as saying
that S ∩ T = Φ. We say that S and T are overlapping if there are elements in S ∩ T .
#(S ) is the number of elements that are in S . If S is infinite, we say #(S ) = ∞; otherwise, #(S ) is a non-negative
integer.
For any two sets S and T :
#S ∪ T ) = #(S ) + #(T ) − #(S ∩ T ) ;
Venn Diagram. This is an image of the relationships among various substes of the Universal Set involved in the
problem. Represent the entire universe U by a rectangle, and draw a circle for each of the subsets of interest. In
general, with no given information, draw the figure so that all circles intersect with each other. If the information
is that two particular subsets do not intersect, move the image so that the corresponding circles do not overlap. If
one subset is to be contained in another,move the corresponding circles so that one is contained in the other.
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