Rules of traversing the Ghost Leg ◦  Pick a start point at the top ◦  Follow down the ladder   If you reach a horizontal “leg”, you must take it   Continue following down

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

  Figure out all the ghost legs with two vertical lines.

 You can add as many horizontal lines as you prefer.

 What do you notice?

 Before, we looked at permutations of the set of two elements: {1, 2}

 Now let’s look at permutations of the set of three elements: {1, 2, 3} ◦ Draw out all the possible ghost legs that

rearrange the set ◦ How many are there? How do you know you

have them all??

 A permutation is basically an arrangement of a set ◦  In concrete mathematical terms, a

permutation is a one-to-one and onto mapping from a set to itself.

 A ghost leg is just a way to represent permutations that is easy to relate to.

 An operation is a way to get new elements from existing elements. ◦  Example: Addition, subtraction, multiplication,

division are operations that deal with numbers

 Can we define any operations with ghost legs?

 Ghost Legs are forms of functions ◦  For a particular input, we have an output

 We can compose ghost legs just as we compose functions ◦ We traverse through one ghost leg and then

we traverse through the other one thus resulting in another ghost leg.

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

1 2 3

 Put the above example in the “number/arrow” form instead of ghost leg

  Is there anything interesting that you find??

 Work on Handout #2

  Inverse Ghost Legs   Identity Ghost Leg  Mathematical term for this: Permutation

Groups ◦  Binary Operation ◦  Identity ◦  Inverse

  If we have time…Composition of more than two ghost legs ◦  Associative?? ◦  Commutative??