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MTH110

Assignment 1

Ryerson University
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Sets and Statements in Tilomino

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Due: November 5 in Class


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Rules

  • Please read the MTH110 General Assignment Information before submitting your assignment.

    Preamble

    This assignment is based on Tilomino, also known as Tarski's world. It is 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. There is also information about Tilomino in the textbook under the name Tarski's world, see pages 85, 98, 99, 103-107 and 120-121.

    Additional notation will need to be introduced to describe this assignment and the next one. This notation is described in a separate web page titled Additional Tilomino Notation for Assignments.

    Part 1 - Propositional Calculus

    1.1 From English to Tilomino

    For each of the English statements below:
    1. Translate the statement from English into Tilomino.
    2. Use the Tilomino program to evaluate the Tilomino translation and briefly explain the answer it gives.
    Statements:

    1. In the Two by Two world:
      b and c are the same size and shape, but a is different in both attributes.

    2. In the The World is Round world:
      b is below a, but neither a nor c are.

    3. In the Four Squares world:
      b is a large circle only if a is a small square.

    4. In the Smiling World world:
      e and h being in the same column is a necessary condition for e to be a small circle.

    5. In the All Here world:
      c being a small triangle to the left of d is not a sufficient condition for a to be a large square below d


    Part 2 - Sets

    2.1 Set Interpretation

    For each of the sets at the end of this question,
    1. Explain in English what the set is.
    2. Where possible give an exhaustive definition of the set (listing all the elements inside curly brackets).

    Note that your English explanation should not contain references to quantifiers or free variables, i.e. the variables (like x, y, z, w) that follow quantifiers.

    Example
    For example, for the set The answer would be:
    1. The set of all worlds in which the tile 'a' is a square.
    2. EXAMPLE = { F, A, B, H }
    Here are the sets:

    1. A = {wWORLDS | ∃ x ∈ w such that small(x)},

    2. B = { x ∈ TILENAMES | ∃ w ∈ WORLDS, x ∈ w ∧ small(x) }

    3. L = {wWORLDS | ∀ x ∈ w, circle(x)}.

    4. M = {wWORLDS | ∃ s ∈ SHAPES such that ∀ xw, ~s(x) }

    5. N = {sSHAPES | ∃ w ∈ WORLDS such that ∀ x, yw, s(x) = s(y) }

    2.2 Set Operations

    1. For a given world w ∈ TILOMINOUNIVERSE and x ∈ TILENAMES define

      Tx(w) = { y ∈ w | y has label x }.

      Let S(w) = { Tx(w) | x ∈ TILENAMES } - { φ },

      1. List the elements of S(2)

      2. Is { Tx(w) | x ∈ TILENAMES } - { φ }, a partition of w for every w ∈ TILOMINOUNIVERSE,? Explain your answer.

    2. Find the following:
      1. ℘(S) where S = { x ∈ 2 | COLOF(x, R) ≤ 7 } (℘(A) is the power set of A).

      2. R2,
      3. R × 2.

    Part 3 - Predicate Calculus - Quantifiers

    3.1 From English to Tilomino

    For each of the English statements below:
    1. Translate the statement from English into Tilomino.
    2. Use the Tilomino program to evaluate the Tilomino translation for each world given.
    3. In each case give a short explanation of the answer it gives.
    Example:

    For example if the statement is:

    The answer might look like:

    Statements:

    1. The southernmost block is a square.
      1. Smiling World
      2. All Here Revisited

    2. All tiles except circles are medium squares.
      1. Smiling World
      2. The World is Round

    3. Whenever a circle is east of a square there is a triangle west of that circle.
      1. Two by Two
      2. All Here Revisited

    4. For every size that appears, there are at least two blocks of that size.
      1. Four Squares
      2. All Here

    5. For a block to be round it is not sufficient that it be West of a triangle.
      1. All Here
      2. The World is Round

    3.2 Translate expressions

    Translate each of the Tilomino statements below into English. This translation should not contain references to quantifiers or free variables, i.e. the variables (like x, y, z, w) that follow quantifiers.

    Where relevant the statements talk about a world N. You will need to use the Tilomino Notation to understand the some of the statements. Many will not work in the Tilomino Program, though some will.

    1. ∀ y ∈ LABELS, ∃ x ∈ N, ROW(x, N) = y;

    2. ∀ x ∈ N, ∃ y ∈ N, x ≠ y ∧ samecol(x, y) ∧ (∀ z ∈ N (z ≠ x ∧ z ≠ y) → ~samecol(x, z))

    3. ∀ x, y ∈ TILENAMES, (x ∈ N ∧ y ∈ N ∧ larger(x, y)) → x > y

    4. ∀ x, z ∈ N, (x ∈ TILENAMES ∧ z ∈ TILENAMES ∧ x > z) → (∀ y ∈ TILENAMES, x > y > z → y ∈ N)

    5. ∀ x, y ∈ N, COLOF(x, N) - COLOF(y, N) ∉ {-1, 1}

    6. (∀ x ∈ SHAPE, ∃ y ∈ N, x(y)) ∧ (∀ x ∈ SIZE, ∃ y ∈ N, x(y))

    7. ∀ x, y ∈ N (samecol(x, y) ∧ larger(x, y)) → (square(x) ∧ triangle(y) ∧ above(x, y))

    8. ∀ x, y ∈ TILENAMES (x ∈ N ∧ y ∈ N ∧ x > y) → (northof(x, y) ∧ (sameshape(x,y) → rightof(x, y)))

    9. ∀ x, y ∈ N, circle(x) → (rightof(x, y) ∨ samerow(x, y))

    10. COL(a, N) = 2 ∧ ~samecol(g, h) ∧ samecol(a, f)


    Hand in

    A writeup of your answers to all questions. The Assignment 1 Marking Sheet (in pdf format) stapled to the front of your assignment


    Maintained by Peter Danziger.
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    Last modified Thursday, 12-Nov-2009 06:52:56 EST