COM-FSM MS 095 PreAlgebra Cognitive Approach: Subsets to Pascal
Set Theory
A set is a group formed by a classification, a
well-defined collection of objects.
The members of a set are called the elements of the set.
Here we are using the English meaning of set as a grouping: a set of spoons, a dining
room set.
Well-defined sets:
The set of all islands in Micronesia
The set of all atolls in Yap State
The set of all cars on Mokil
The set of all integers
Not well defined:
The set of the best places to live in the world
The set of all beautiful women in Nigeria
The set of interesting numbers
All of these are vaguely defined because they involve opinion and judgement.
Venn Diagrams
Venn diagrams are pictures of sets. Venn diagrams were named for the mathematician John Venn.
The Universal set is the set of all elements under consideration. The Universal set in a Venn diagram can drawn as a rectangle with the elements drawn inside the rectangle.
When written out on a piece of paper sets are enclosed using bracket symbols: { }
The set could be written U = {marble, ping pong, tennis, superball, wiffel } with each element separated by a comma from the other elements. Often lower case letters are used as abbreviations for elements of a set:
U = {m, p, t, s, w}
There is a special symbol for a set that has no
elements. A set with no elements is called an empty set. The symbol
for an empty set is Ø. This is also sometime written {}.
The set of all cars in Mokil is an empty set - there are no cars in Mokil.
This would be written: Ø
Subsets
Sets made up of some or all of the elements in the Universal set are called subsets. On a large rectangular table is a collection of balls and marbles.
Let U = {spheres on the table}
The ping pong ball and golf ball would be a subset of this Universal set. This would be written:
{ping pong ball, golf ball} Í U
or
{ping pong ball, golf ball} Í {spheres on the table}
In a Venn diagram a circle would be drawn around the ping pong ball and the golf ball, in class a circle of yarn is placed around the balls for each example and a diagram is drawn on the board.
Sample board diagram: ping pong ball subset.
Another subset would be the set of all white balls.
Another subset would be the set of all hollow balls.
Anything a yarn circle can be placed around is a subset:
That is, a subset can be a smaller set or an equivalent set to the larger set, hence the symbol . The "larger" set is sometimes called the superset.
A set which does not contain all of the members of the larger superset is called a proper subset. The set of ping pong balls is a proper subset. The set of white balls is a proper subset. The set of hollow balls is a proper subset.
The symbol is used for a proper subset: Ì
Let U = {spheres on the table}
{white balls} Ì U
{hollow balls} Ì U
{balls that are used in sports that require a paddle or racket} Ì U
What is the set of all basketballs on the table? There are no basketballs, thus the set is an empty set, Ø. This would be represented by placing a circle around none of the balls on the table. The empty set is a subset of all sets.
Numbers of Subsets
Consider an empty set. How many subsets are there of the empty set?
One. All sets include an empty set subset, even the empty set.
The empty or null set has only itself as a subset:
Ø Í Ø 1 empty set subset
Consider a set containing a single tennis ball {t}.
How many subsets does it have?
A Universal set with one element {a tennis ball} has how many empty set
subsets?
How many different objects can a single yarn circle surround in a single
element Universal set?
U = {t}
Ø Í {t} 1 empty set subset
{t} Í {t} 1 single element subset
2 subsets of a 1 element Universal set
A set with two elements, a tennis ball and a ping pong ball, U = {t, p} has:
Ø Í {t,p} 1 empty set subset
{t} Í {t,p} {p} Í {t,p} 2 single element subsets
{t,p} Í {t,p} 1 two element set subset
4 subsets of a 2 element Universal set
A set with three elements, a tennis ball, ping pong ball, and a golf ball, U = {t, p, g} has:
Ø Í {t,p,g} 1 empty set subset
{t} Í {t,p,g} {p} Í {t,p,g} {g} Í {t,p,g} 3 single element subsets
{t,p} Í {t,p,g} {t,g} Í {t,p,g} {p,g} Í {t,p,g} 3 two element subset
{t,p,g} Í {t,p,g} 1 three element subset
8 subsets of a 3 element U
Tackling a set with three elements, a tennis ball, ping pong ball, golf ball, and marble means writing {t, p, g, m} over and over again. Mathematics is, at times, about a peculiar form of laziness that substitutes abbreviations for longer expressions. Here we will write U for the Universal set instead of {t,p,g,m}. Here too is a good juncture to get the students working at their desks and trying to work this one out. I have in the past introduced the above three cases as lecture/demonstration. This provides a foundation for exploring the more complex four element case.
U = {t,p,g,m}
Ø Í U 1 empty set subset
{t} Í U {p} Í U {g} Í U {m} Í U 4 single element subsets
{t,p} Í U {t,g} Í U {t,m} Í U
{p,g} Í U {p,m} Í U {g,m} Í U 6 two element subsets*
{p,g,m} Í U {t,g,m} Í U {t,p,m} Í U {t,p,g} Í U 4 three element subset**
{t,p,g,m} Í U 1 four element subset
16 subsets of a 3 element U
* Form these by "rotating" three elements
against one and then placing the one outside. In this case the tennis ball is paired with
the other three and then set aside. That accounts for all tennis ball cases.
** Form these triples by considering the elements that are not in
the corresponding single element subsets above.
After thrashing through the above I usually then ask, well how about a five element Universal set? How many three element subsets? Is there anyway to predict or shortcut the lengthy (and error prone) trial and error? In one class I had the class begin to work out the five element Universal set at their desk and I assigned the five and six element Universal set as homework, picking up from this point at the next lecture. I wanted them to struggle with the difficulty of the task so that they would actually appreciate any shortcut we could find.
Consider the sequence:
1 1 1 1 1 1
1 2 3 4 5
1 3 6 10
1 4 10
1 5
1
_ _ _ _ __ __
1 2 4 8 16 32
Turn the sequence sideways to get:
1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1
And then spread it out as shown below:
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
Can anyone see a way to calculate one row from the
previous row?
The pyramid is called Pascal's Triangle.
Add each row of Pascal's Triangle:
1 = _____
1 + 1 = _____
1 + 2 + 1 = _____
1 + 3 + 3 + 1 = _____
1 + 4 + 6 + 4 + 1 = _____
1 + 5 + 10 + 10 + 5 + 1 = _____
Use your head, or paper and pencil, or a calculator to calculate the following results. On a scientific calculator the yx key or the xy key does exponents.
22 = 2 × 2 = _____
23 = 2 × 2 × 2 = _____
24 = 2 × 2 × 2 × 2 = _____
25 = 2 × 2 × 2 × 2 × 2 = _____
Match these results to those for the rows of Pascal's
triangle.
Use your Pascal's triangle results to calculate the next two results:
20 = _____
21 = _____
Observing the following pattern:
24 = 2 × 2 × 2 × 2
23 = 2 × 2 × 2
22 = 2 × 2
Write the last two rows based on the pattern:
24 = 2 × 2 × 2 × 2
23 = 2 × 2 × 2
22 = 2 × 2
21 = _____
20 = _____
Pascal's Triangle appears in many places. Use the yx key or the xy key on your calculator to do the following:
110 =
111 =
112 =
113 =
114 =
On a Sharp EL-506L Advanced D.A.L. calculator (not on a Sharp EL-531 DAL!) switch to base 16 hexidecimal and you can get: 15AA51
This page hosted by 
Get your own Free Home Page