MTH110 |
Assignment 1 Solutions | |
Sets and Statements in Tilomino |
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This solution page is meant to give you an idea of the correct answers to the assignment questions. It is not necessarily comprehensive.
It should be noted that there are often different forms of the correct answer.
In this regard note that the following pairs of functions are equivalent:
(Eastof, Rightof), (Westof, Leftof),
(Northof, Above), (Southof, Below)
On the other hand note that ~Northof is not equivalent to Southof (since they may be on the same row) and so on.
In many cases DeMorgan's, or other rules, give a different form for the answer.
Sameshape(b,c) & SameSize(b,c) & ~SameSize(a,b) & ~SameShape(a,b)
≡ Sameshape(b,c) & SameSize(b,c) & ~SameSize(a,b) & ~SameShape(a,b)
& ~SameSize(a,c) & ~SameShape(a,c)
True: b and c are both large squares, but a is a small triangle.
below(b,a) & ~(below(a,a) | below(c,a))
≡ below(b,a) & ~below(a,a) & ~below(c,a)
False: c is below a.
(large(b) & circle(b)) -> (small(a) & square(a))
True(!): b is not a circle, so the predicate is false and thus the implicative statement is true.
~(samecol(e,h)) -> ~(small(e) & circle(e)) ≡ (small(e) & circle(e)) -> samecol(e,h)
True: The predicate is false.
~((small(c) & triangle(c) & leftof(c,d)) -> (large(a) & square(a) & below(a,d)))
≡ (small(c) & triangle(c) & leftof(c,d)) & ~(large(a) & square(a) & below(a,d)))
≡ (small(c) & triangle(c) & leftof(c,d)) & (~large(a) | ~square(a) | ~below(a,d))
False: c is not a small triangle.
Note that, as above, your English explanation should not contain references to quantifiers or free variables, i.e. the variables (like x, y, z, w) that follow quantifiers.
{ {a}, {b}, {c} }
No. Not all tiles are labeled so the union is not all of the set.
{ ∅, {a}, {b}, {c}, {a,b}, {b,c}, {a,c}, {a,b,c} }
{[R:a], [R:b], [R:c], [R:(8,6)], [2:a], [2:b], [2:c], [2:(4,8)]}
{([R:a], [2:a]), ([R:a], [2:b]), ([R:a], [2:c]), ([R:a], [2:(4,8)]), ([R:b], [2:a]), ([R:b], [2:b]), ([R:b], [2:c]), ([R:b], [2:(4,8)]), ([R:c], [2:a]), ([R:c], [2:b]), ([R:c], [2:c]), ([R:c], [2:(4,8)]), ([R:(8,6)], [2:a]), ([R:(8,6)], [2:b]), ([R:(8,6)], [2:c]), ([R:(8,6)], [2:(4,8)]) }
Ex square(x) & Ay x=y | southof(x,y)
False: There is no southernmost block, The two squares are in the same row.
True: a is a square south of every other block.
Ax (medium(x) & square(x)) | circle(x)
≡ Ax ~(medium(x) & square(x)) -> circle(x)
≡ Ax ~circle(x) -> (medium(x) & square(x))
True: The circles a and e are the only tiles that are not medium squares.
True: All blocks are circles, so the statement is vacuously true.
Ax (circle(x) & Ey (square(y) & eastof(x,y))) -> Ez (triangle(z) & westof(z,x))
True: There are no circles in this world, so the statement is vacuously true.
True: Note that the circle in (2,2) is not east of any squares.
AxEy x#y & (small(x) -> small(y)) & (medium(x) -> medium(y)) & (large(x) -> large(y))
False: a is the only small block.
The statement is true.
~(Ax (Ey triangle(y) & westof(x,y)) -> circle(x))
≡
~(Ax Ay (triangle(y) & westof(x,y)) -> circle(x))
≡
Ex Ey triangle(y) & westof(x,y) & ~circle(x)
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.
Note that "below" does not necessarily mean in the same column. You may use the phrase "directly below" to mean "below and in the same column" (below(x,y) & samecol(x,y)). Similarly, you may use the phrase "directly above" to mean "above and in the same column" (above(x,y) & samecol(x,y)).
a is in column 2 and so is f, but g and h are in different columns.
Maintained by Peter Danziger.
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Last modified
Friday, 13-Nov-2009 09:45:17 EST