MTH110
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Additional Tilomino Notation for Assignments
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Table of Contents
Introduction and Page Guide
MTH110 assignments are based on
Tilomino, a universe consisting of tiles of various sizes, shapes and colors,
about which logical questions can be asked. Tilomino can be accessed
by any Java enabled browser on any modern operating system.
Note that you will be expected to be able to answer questions on
Tilomino in the midterm and final exams.
The sections on this page are in order.
This means that in order to understand a specific section, you will need
to read the sections that precede it.
In some of the assignment questions
you will be asked to translate statements to
or from "Tilomino". By this, we mean that you may use any of the functions,
logical connectives, and quantifiers defined in Tilomino. Alternatively
you may use the standard logical symbols for connectives and quantifiers
if you wish. However, if you wish to input a statement into Tilomino you
must use the syntax of the Tilomino program.
Additional Tile Notation
Here is some additional notation for tiles that you will need
for your assignments.
This notation will not work on the version of Tilomino.
- Simple Notation for Unnamed Tiles
If, in your assignments, you need to refer to a tile which does not have
a name, you may reference it by (row, column), starting from (1,1) in the
top left corner.
For example, if you need to discuss the unnamed circle in the
The World is Round world
you can refer to it as (6,8).
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
- Generalized Tile Notation
In many of the assignment questions, you will only work on one world at
a time, so simple tile notation (e.g. a, b, c, (6,8))
will be enough.
However, in some of the questions we will need to discuss tiles in more than one world.
This will be problematic because the same tile names (e.g. a, b, c)
are used from one world to another even though
tiles of the same name in different worlds are not the same.
For example even though
the All Here world has a large square called a in position (8,8) and
the All Here Revisited world also has a large square called a in that
same position,
these two tiles are not the same, simply because they are in different worlds.
Therefore, if we want to discuss a collection of tiles from different worlds,
we need a more precise naming mechanism for tiles that
not only names the tile that is being discussed but also the world containing this tile.
The notation used is:
[world:tile] which refers to a specific tile in a specific world.
For example:
[R:a] is tile a in world R,
[S:(7,4)] refers to the tile on cell (7,4) in the Smiling
World.
For the earlier example above,
[A:a] refers to tile a in the All Here world,
whereas
[H:a] refers to tile a in the All Here Revisited world.
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
Set Notation and Sets
Set theory notation will be used throughout the assignments.
The following notation will be used in the description of the assignments.
You can (and should) use this notation in your assignment writeups, but you will not
be able to use it with Tilomino.
For a general introduction to the set theory concepts needed to understand
this section, please read section 5.1 of the Epp textbook pages 255 to 262.
- Set Notation for Tilomino Worlds
A Tilomino world is really a set of tiles.
Therefore we will use the name of a world to refer to the set of tiles in that world.
For example, 2 = Two By Two = {a, b, c, (8,4)}
Of course, worlds can also be defined using the generalized tile notation:
For example, 2 = Two By Two =
{ [2:a], [2:b], [2:c], [2:(8,4)] }
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
- Other Tilomino Sets
In addition to the Tilomino worlds just described,
some of the assignment questions will refer to the following sets:
- TILOMINOUNIVERSE = {all possible Tilomino worlds}
= {all possible 8X8 grids where some of the cells contain a Tilomino tile}
- WORLDS = {all worlds in the Tilomino program} =
{F, 2, R, A, B, S, H}
- TILENAMES = {names of tiles in Tilomino} = {a, b, c, d, e, f, g, h, i, j}
- SHAPES = {shapes of tiles in Tilomino} = {circle, square, triangle}
- SIZES = {sizes of tiles in Tilomino} = {small, medium, large}
- LABELS = {labels of rows and columns of Tilomino} = {1, 2, 3, 4, 5, 6, 7, 8}
- TILOMINO = {x
w |
w
WORLDS}
- Given any i
LABELS,
and any w
TILOMINOUNIVERSE, define
- ROW(i,w) = {x
w |
x is in the ith row of world w}
- COL(i,w) = {x
w |
x is in the ith column of world w}
- Given any i in LABELS, define
- ROWS(i) = {row(i,w) | w
WORLDS}
- COLS(i) = {column(i,w) | w
WORLDS}
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
Functions and Relations
- Additional Functions
The assignments may also refer to the following functions.
Given any tile x
and any w
TILOMINOUNIVERSE, define
- ROWOF(x,w) =
- 0 if x
w;
- n
LABELS s.t.
x
ROW(n,w) otherwise;
- COLOF(x,w) =
- 0 if x
w;
- n
LABELS s.t.
x
COL(n,w) otherwise;
For example:
ROWOF(e,B)=7 and COLOF(e,B)=3 but
ROWOF(e,2)=0 and COLOF(e,2)=0 because there is no e tile in world 2.
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
- Additional Relations
The assignments may also refer to the following relations:
Remember: this notation is only for your assignment description and writeup.
It will not work in Tilomino.
Maintained by Peter Danziger.
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Last modified
Friday, 07-Oct-2005 09:51:47 EDT