Re: [Haskell-cafe] Learn You a Haskell for Great Good - a few doubts

On 4/03/2011, at 10:47 PM, Karthick Gururaj wrote:

On Fri, Mar 4, 2011 at 10:42 AM, Richard O'Keefe wrote:

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On 4/03/2011, at 5:49 PM, Karthick Gururaj wrote:

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I meant: there is no reasonable way of ordering tuples, let alone enum them.

There are several reasonable ways to order tuples.

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That does not mean we can't define them: 1. (a,b) > (c,d) if a>c

Not really reasonable because it isn't compatible with equality.

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2. (a,b) > (c,d) if b>d 3. (a,b) > (c,d) if a^2 + b^2 > c^2 + d^2 4. (a,b) > (c,d) if a*b > c*d

Ord has to be compatible with Eq, and none of these are.

Hmm.. not true. Can you explain what do you mean by "compatibility"?

Trichotomy. Ad definition 1: (1,2) > (1,3) is false (1,2) > (1,3) is false but (1,2) ==(1,3) is also false. Ad definition 2: Same as definition 1 only flipped. Ad definition 3: (3,4) > (4,3) is false (3,4) < (4,3) is false but (3,4) ==(4,3) is also false. Ad definition 4: (1,0) > (0,1) is false (1,0) < (0,1) is false but (1,0) ==(0,1) is also false. Ord is a subclass of Eq, after all.

One of the following, and exactly one, must always hold, on a ordered set (is this what you mean by "compatibility"?), for any arbitrary legal selection of a, b, c and d. a. (a, b) EQ (c, d) b. (a, b) LT (c, d) c. (a, b) GT (c, d)

All the definitions above are compatible in this sense.

As I just showed, none of them is.

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As a side note, the cardinality of rational numbers is the same as those of integers - so both are "equally" infinite.

Ah, here we come across the distinction between cardinals and ordinals. Two sets can have the same cardinality but not be the same order type. (Add 1 to the first infinite cardinal and you get the same cardinal back; add 1 to the first infinite ordinal and you don't get the same ordinal back.)

:) Ok. The original point was whether there is a reasonable way to enumerate a tuple, I guess there is none.

And the original point was about ordering, which is why it's relevant that there are more order types than size types.