How Many Subsets And Making Connections

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How Many Subsets?AndMaking Connections
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A set is simply a collection of distinct objects.
Suppose a band had Jean, Juana, Sean, Jack and
Ivan as members. We could talk about the set B
Jean, Juana, Sean, Jack, Ivan
A subset of B would be any set all of whose
members are also members of Set B, that is
members of the band.
Notice that a set is a subset of itself since all
its members are members of itself! The empty
set is considered a subset of any set. You
must admit that the empty set has no members that
are not members of the original set.
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Lets look at the same thing with Venn diagrams,
Again Set B Jean, Juana, Sean, Jack,
Ivan Lets have Set C Jean, Juana,
Sean Lets have Set D Ann, Dan
Set C is a Subset of Set B, because all of the
members in Set C are also in Set B.
Sets B and D are disjoint, because they have no
members in common.
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This activity is meant to help you develop a
systematic way to list subsets and, through
noticing some patterns, make connections with
other mathematics that you are familiar with.
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Start by listing the subsets of each of the four
sets below. Please write your subsets down in an
organized fashion so that you can refer to them
later.
A
A, T
T, E, A
H, E, A, T
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Here are the subsets of the first three sets.
Check your subsets. If you are missing any,
consider how you might become more accurate in
your listings. Being systematic is essential as
sets get larger.
Sets Subsets
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One Systematic Way to List all the Subsets of
H, E, A, T
Sets with Zero Members
Sets with One Member

H E A T
Sets with Two Members
Sets with Three Members
H,E H,A H,T E,A E,T
A,T
H,E,A H,E,T H,A,T E,A,T
Sets with Four Members
H,E,A,T
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The set T, E, A has three members and eight
subsets. The set 5, 6 ,7 would also have
eight subsets. Make sure that you can see that
any three member set would have eight subsets.
Using the four sets that you found subsets for,
make a table with one column noting the number of
members a set has (1, 2, 3, 4) and the next
column noting the number of subsets. Look for
patterns and determine how many subsets a five
member set would have.
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Here is my table
What patterns do you see? Given the number of
sets come up with an algebraic expression that
will give the number of subsets.
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You should notice that the number of subsets
doubles each time the number of members increases
by one. Algebraically the number of subsets is
2n where n is the number of members of the set.
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How many subsets does T, E, A, C, H have?
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You probably noticed that the four sets we have
used, A, A, T, T, E, A and H, E, A, T a
new letter is added for each new set. The next
slide shows how adding one new member doubles the
number of subsets.
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1
A
A

2
A,T
T
A
4
A,T
A T A,T
T,E,A
8
E
A,E
T,E
A,T,E
A T A,T E A,E T,E A,T,E
H,E,A,T
H A,H T,H A,T,H E,H A,E,H
T,E,H A,T,E,H
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Did you notice that all the subsets of A, T are
also subsets of T, E, A. These are , A,
T and A, T. The additional subsets of T, E,
A are these same subsets with an E added. E,
A, E, T, E and A, T, E. This explains why
the number of subsets double when the number of
members increases by one.
You may want to go back and notice the same thing
going from T, E, A to H, E, A, T. Then use
what you have learned to list all the members of
the set T, E, A, C, H.
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H,E,A,T
H E A T
H,E H,A H,T E,A E,T A,T
H,E,A H,E,T H,A,T E,A,T
H,E,A,T
16 SUBSETS
H E A T
H,E H,A H,T E,A E,T A,T
H,E,A H,E,T H,A,T E,A,T
H,E,A,T
T,E,A,C,H
32 SUBSETS
C H,C E,C A,C T,C H,E,C
H,A,C H,T,C E,A,C E,T,C A,T,C H,E,A,C
H,E,T,C H,A,T,C E,A,T,C
H,E,A,T,C
We start with all the subsets we already had.
We add a C to each of these sets doubling the
number of subsets.
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We will look at one more pattern and then relate
it to something from Section 1.3 of your text as
well as something from College Algebra. Start by
filling in the following tables.
What do the numbers remind you of? If you cant
figure it out reread Investigation 1.5 in your
text.
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We have connected listing subsets to Pascals
Triangle, two seemingly unrelated mathematical
concepts. This is what Bassarear was talking
about in the last section of Chapter 1. The more
connections you as a teacher can make for your
students, or better yet, have your students
discover on their own, the more likely your
students will enjoy and retain their learning.
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Here are my tables
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One more connection and we will be through. In
College Algebra you were asked to expand binomial
expressions such as (x 2)4 using Pascals
Triangle.
You started with the xs and the 2s.
x4 x3 x2 x1
x0
20 1 21 2 22 4 23
8 24 16
Note the pattern in the exponents 4, 3, 2, 1, 0
and 0, 1, 2, 3, 4. This relates to the 4 member,
3 member, 2 member, 1 member and 0 member subsets
found in the set H, E, A, T.
Next a row of Pascals triangle was applied and
the three parts were multiplied to form each term
of the expanded expression.
1 4 6
4 1
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x4 8x3 24x2
32 x 16
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Dont worry, you will not be asked to expand a
binomial on the test. However you need to be
able to systematically list the subsets of a set
of up to five members, construct Pascals
triangle and use Pascals triangle to determine
the number of subsets with a particular number of
members.
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return.