Set Theory and Pascal

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:

Not well defined:

All of these are vaguely defined because they involve opinion and judgement.

Venn Diagrams

Venn diagrams can assist in the analysis 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 is drawn as a rectangle. A Universal set consisting of all the spheres on the table would be drawn:

Set of spheres

Sets are enclosed using bracket symbols: { }

The set could be written U = {marble, ping pong, tennis, superball, wiffle}

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 .

The set of all cars in Mwoakil is an empty set: there are no cars in Mwoakil.

This would be written: {}

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 = {x|x is a sphere on the table}

The ping pong balls would be a subset of this Universal set. This would be written:

{ping pong balls} U or {ping pong balls} {spheres on the table}

In a Venn diagram a circle would be drawn around the ping pong balls, in class a circle of yarn is placed around the balls for each example and a diagram is drawn on the board.

Illustration of a subset

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.

The set of all the balls would also be a subset of the balls. 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.

Let U = {all spheres on the table}

{ping pong balls} U
{white balls} U
{hollow balls} U

What is the set of all basketballs on the table? There are no basketballs, thus the set is an empty set, {}. 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.

Add children's letter cubes to the table:

Note that {letter cubes} {spheres on the table} because cubes are not spheres.

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:

{ }  { }	1 empty set subset

Consider a set containing a single tennis ball. How many subsets does it have?

A Universal set with one element (a tennis ball) has:

U = {t}

{ }  {t}	1 null set subset
{t}  {t}	1 single element subset
                2 subsets of a 1 member universal set.

A set with two elements, a tennis ball and a ping pong ball, U = {t, p} has:

{ }    {t,p}        1 null set subset
{t}    {t,p} 
{p}    {t,p}        2 single element subsets
{t,p}  {t,p}        1 two element subset
                     4 subsets of a 2 element universal set 

Work in a group with three different unique objects to find the number of subsets for a set of three objects.

_____ How many empty set subsets?
_____ How many single element subsets?
_____ How many different and unique two element subsets?
_____ How many three element subsets?
_____ How many total subsets?

Class-wide results will be posted to the board.

Work in a group with four different unique objects to find the number of subsets for a set of three objects.

_____ How many empty set subsets?
_____ How many single element subsets?
_____ How many different and unique two element subsets?
_____ How many different and unique three element subsets?
_____ How many four element subsets?
_____ How many total subsets?

Class-wide results will be posted to the board.

Work in a group with five different unique objects to find the number of subsets for a set of three objects.

_____ How many empty set subsets?
_____ How many single element subsets?
_____ How many different and unique two element subsets?
_____ How many different and unique three element subsets?
         (Hint: Every time you made a two element subset the left over objects formed a unique three element subset)
_____ How many different and unique four element subsets?
_____ How many five element subsets?
_____ How many total subsets?

Class-wide results will be posted to the board.

How many total subsets are there for a six element set?

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

The pyramid is called Pascal's Triangle.

Pascal's Triangle

Add each row of Pascal's Triangle:

1 = 1
1 + 1 = ___
1 + 2 + 1 = ___
1 + 3 + 3 + 1 = ___
1 + 4 + 6 + 4 + 1 = ___
1 + 5 + 10 + 10 + 5 + 1 = ___

Use a calculator to calculate the following results. On a scientific calculator the yx key or the xy key does exponents.

20 = ___
21 = ___
22 = 2 2 = ___
23 = 2 2 2 = ___
24 = 2 2 2 2= ___
25 = 2 2 2 2 2 = ___

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 =

Calculate the following:

(x + 1)(x + 1) =

(x + 1)(x + 1)(x + 1) =
Do the above by multiplying (x + 1)(x + 1) and then multiplying that result by (x + 1)

(x + 1)(x + 1)(x + 1)(x + 1) =

(x + 1)(x + 1)(x + 1)(x + 1)(x + 1) =