Maths trivia

[Joe4]

Just had one thought here before I go to bed.

Is my problem that I am trying to apply concepts on finite set logic to infinite sets? Dealing with infinite sets kind of changes the rules and logic a bit.

 

[Joe4]

Here's one that is more in my "wheelhouse". Many people have a hard-time understanding the Monty Hall Paradox, and the concept that given this scenario, you should always switch your choice (to increase your odds of winning).

 

[pgc01]

The way I see it, you have gone through 6 integers so far, and come up with 17 rational number. So I guess I am not understanding the 1-1 relationship.

If you imagine extending the table to infinity, both in the number of rows and in the number of columns, you could always

- given a whatever natural number n, there's 1 (and 1 only) corresponding rational number (the one in column B, row n)

- given a whatever rational number n, there's 1 (and 1 only) corresponding natural number (we look up the rational number in column B and get the corresponding natural number in column A)


Notice that it does not mean that by ordering the sets differently you cannot get to other conclusions. In your example it seems that there are more rationals, in Oaktree's example it seems there are more integers. In my other example with Integers and Naturals with 2 different ways of ordering the sets I got 2 different conclusions.

This is usual in infinite sets. If you order them differently you can get contradictory conclusions. The rule is then to consider the sets to have the same cardinality if you can find at least 1 way to order the sets so that they have a bijective relationship, meaning a 1 to 1 relationship in both senses.

So, in this case, since we can find at least 1 way to order the sets (the one I posted in step 3) that establishes a correspondence 1 to 1 in both senses betweeen the 2 sets, we say they have the same cardinality.

 

[pgc01]

On a lighter note, when you add odd numbers, starting by 1, you always get squares:

1 = 1²
1 + 3 = 2²
1 + 3 + 5 = 3²
1 + 3 + 5 +7 = 4²
1 + 3 + 5 +7 + 9 = 5²
...

 

[Expiry]

On a lighter note, when you add odd numbers, starting by 1, you always get squares:

1 = 1²
1 + 3 = 2²
1 + 3 + 5 = 3²
1 + 3 + 5 +7 = 4²
1 + 3 + 5 +7 + 9 = 5²
...

Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.

 

[schielrn]

Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.

 

[Joe4]

Another way to always get squares, is to open up a discussion on Maths Trivia. Ho ho ho.

As Huey Lewis used to say, "its hip to be square"!

 

[Oaktree]

I have no doubt that you're right, Pedro.

But this:

That's why in mathematics, when you are working with infinite sets you must use other rules.

is when I realized I didn't want to go further down the path of obscurity that is higher level math. IMO, math divorced from logic leads to crazy people.

The set of rationals includes integers + other things, but the set of rationals is the same size as the set of integers. That makes more sense

 

[Joe4]

is when I realized I didn't want to go further down the path of obscurity that is higher level math.

Amen to that, brother! I was a math major myself in college, but when you get into some of that "high" level stuff like dealing with infinite sets, it can really blow your mind! My "logical" mind doesn't like that.

Of course, it has also been 16 years since I looked at anything like this, so I have probably forgotten most of what I learned anyway!

 

[erik.van.geit]

Monty Hall Paradox spreadsheet

Copy down as much rows as you like.
This result was obtained using 24.000 rows.

   A   B          C                 D              E      F    G               
 1 car 1st choice remaining goat(s) opened by host switch win? probability win 
 2 2   1          3                 3              2      1    65,81999%       
 3 3   3          12                2              1      0                    
 4 2   1          3                 3              2      1                    
 5 3   1          2                 2              3      1                    
 6 2   2          13                1              3      0                    
 7 2   3          1                 1              2      1                    
 8 2   3          1                 1              2      1                    
 9 1   1          23                3              2      0                    
10 2   1          3                 3              2      1                    
Blad1
[Table-It] version 09 by Erik Van Geit

RANGE   FORMULA (1st cell)
A2:B10  =RANDBETWEEN(1,3)
C2:C10  =SUBSTITUTE(SUBSTITUTE("123",A2,""),B2,"")
D2:D10  =IF(LEN(C2)=1,C2,MID(C2,RANDBETWEEN(1,2),1))
E2:E10  =--SUBSTITUTE(SUBSTITUTE(123,B2,),D2,"")
F2:F10  =IF(A2=E2,1,0)
G2      =SUM(F:F)/(COUNTA(F:F)-1)
[Table-It] version 09 by Erik Van Geit

kind regards,
Erik