Proofs and Logic Numerically Equivalent

TARAJS · 2011-12-01 02:22:47

I need to show how the Rationals (R) and the complex numbers (C) are logically equivalent. or in turn lCl=lR^2l

Help would be amazing! thank you!

John E. Franklin · 2011-12-01 11:53:16

Aren't the rationals displayable on a number line, however the complex numbers
may need a plane if you use the real and imaginary parts as x and y components?
That's all I know of this.

Bob · 2011-12-01 14:28:48

hi TARAJS

I need to show how the Rationals (R) and the complex numbers (C) are logically equivalent. or in turn lCl=lR^2l

Sorry, but this isn't making sense to me.

Sets of numbers are not statements so they cannot be logically equivalent.

http://en.wikipedia.org/wiki/Logical_equivalence

And |C| means the absolute value of C. OK if C were a single value, but C is a set.

Please would you give more detail of how this question has arisen. What are you studying? Are you able to express your problem in a different way?

Bob

VasiliY_Honda · 2011-12-01 14:53:56

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VasiliY_Honda · 2011-12-01 14:55:26

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TheDude · 2011-12-01 15:21:11

I think he's asking about the cardinality of R versus the cardinality of C.

TARAJS · 2011-12-02 00:50:50

I'm in a Proofs and Logic class in college. I'm in the chapter/section on cardinality of sets/ uncountable sets. Previously in the lesson we learned that lrational numbersl = lnatural numbersl (and yes that does mean cardinality). My professor is giving us extra credit if we can find how the rationals are numerically equivalent to the complex numbers. when you put something into the form lxl=lyl it means that x is numerically equivalent to y. it means that every x in the set/subset has a matching unit in y. The answer to the actual statement is yes they are logically equivalent i just need to show how.

TheDude · 2011-12-02 01:21:27

The problem is that the cardinalities of the rationals the complex numbers are not equal. The rationals are countable and the complex numbers are uncountable. Either you're supposed to compare the reals and the complex numbers, or you're using some kind of reduced version of the complex numbers where the real and imaginary parts can only be rationals, or the problem is simply wrong.

careless25 · 2011-12-02 02:37:42

this might help:
http://en.wikipedia.org/wiki/Complex_number#Formal_construction

TARAJS · 2011-12-02 02:52:59

the only other thing he put on the problem was that we can do the rations and complex or that the rationals are numerically equvalent to the rationals squared... is that better?

Bob · 2011-12-02 03:51:20

hi TARAJS

So we are talking about cardinality.

The set of all rationals is countable and has the same cardinality as the integers.

Where you have written down 'rationals' I think you should have said 'reals'.

The set of reals is bigger, and has the same cardinality as the set of all points in a plane and hence the complex numbers.

Start by looking at

http://en.wikipedia.org/wiki/Cardinality

Cardinal arithmetic can be used to show not only that the number of points in a real number line is equal to the number of points in any segment of that line, but that this is equal to the number of points on a plane and, indeed, in any finite-dimensional space.

You can find an outline of how this is proved at

http://en.wikipedia.org/wiki/Space-fill … _of_points

Hope that helps.

Bob

TARAJS · 2011-12-05 04:08:59

i did mean reals! so sorry! and thank you very much for your help!

Math Is Fun Forum

#1 2011-12-01 02:22:47

Proofs and Logic Numerically Equivalent

#2 2011-12-01 11:53:16

Re: Proofs and Logic Numerically Equivalent

#3 2011-12-01 14:28:48

Re: Proofs and Logic Numerically Equivalent

#4 2011-12-01 14:53:56

Re: Proofs and Logic Numerically Equivalent

#5 2011-12-01 14:55:26

Re: Proofs and Logic Numerically Equivalent

#6 2011-12-01 15:21:11

Re: Proofs and Logic Numerically Equivalent

#7 2011-12-02 00:50:50

Re: Proofs and Logic Numerically Equivalent

#8 2011-12-02 01:21:27

Re: Proofs and Logic Numerically Equivalent

#9 2011-12-02 02:37:42

Re: Proofs and Logic Numerically Equivalent

#10 2011-12-02 02:52:59

Re: Proofs and Logic Numerically Equivalent

#11 2011-12-02 03:51:20

Re: Proofs and Logic Numerically Equivalent

#12 2011-12-05 04:08:59

Re: Proofs and Logic Numerically Equivalent

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