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June 18th, 2011, 03:43 AM   #1
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number of elments in integers & rationals

hi all..
i have a question:
how can i prove simply to a 7th grade student that even the integers are subset of the rational numbers, but both sets have the same number of elements?
thank you..
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June 18th, 2011, 02:31 PM   #2
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Re: number of elments in integers & rationals

You could use the standard proof. I'll list the start, with those in [] are omitted, since we count fractions only in lowest terms. The terms are grouped inside {} are those with the numerator + denominator the same.

{1/1},{2/1,1/2},{3/1,[2/2],1/3},{4/1,3/2,2/3,1/4},{5/1,[4/2],[3/3],[2/4],1/5},...
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June 19th, 2011, 08:21 PM   #3
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Re: number of elments in integers & rationals

A seventh-grader is doing cardinality of infinite sets? Whoa. What are they teaching in 7th grade these days?
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June 20th, 2011, 10:48 AM   #4
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Re: number of elments in integers & rationals

mathman... thank you for your help

mathematical.. actually they dont take the cardinality of infinite sets.. but they take the properties of sets in general.. naturals, integers, and rationals.. but some student asked.. if the integers are subset of the rationals, then the number of elements in this set are less?! thats why i needed a simple proof for that.. cause its for a 7th grade student!
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June 27th, 2011, 01:01 AM   #5
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Re: number of elments in integers & rationals

http://en.wikipedia.org/wiki/Galileo%27s_paradox
... In his final scientific work, the Two New Sciences, Galileo Galilei made two apparently contradictory statements about the positive whole numbers. First, some numbers are perfect squares (i.e., the square of some integer, in the following just called a square), while others are not; therefore, all the numbers, including both squares and non-squares, must be more numerous than just the squares. And yet, for every square there is exactly one number that is its square root, and for every number there is exactly one square; hence, there cannot be more of one than of the other. This is an early use, though not the first, of a proof by one-to-one correspondence of infinite sets.
.......
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