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Show that (a,b) + (c,d) = (a+c, b+d) is not well-defined in the rational numbers.

[Note: (a,b) + (c,d) = (ad+bc, bd) is well-defined because (a,b) is related to (c,d) when ad = bc.)]

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- Thread starter iNCREDiBLE
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- #1

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Show that (a,b) + (c,d) = (a+c, b+d) is not well-defined in the rational numbers.

[Note: (a,b) + (c,d) = (ad+bc, bd) is well-defined because (a,b) is related to (c,d) when ad = bc.)]

- #2

Hurkyl

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Have you tried at all? Do you know what it means to fail to be well-defined?

- #3

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Hurkyl said:Have you tried at all? Do you know what it means to fail to be well-defined?

I know what it means. But is it enough with just stating an example?

- #4

HallsofIvy

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What you need to do is find an example of rational numbers (a,b),(a',b'),(c,d),(c',d') such that (a,b) and (a', b') are equivalent, (c,d) and (c',d') are equivalent, but (a+c,b+d) is

(For those who are not clear on this, this is using a method of defining rational number from the integers by saying that two ordered pairs of integers, (a,b) and (a',b') (with second integer non-zero) are equivalent if and only if ab'= a'b. That is an equivalence relation and so partitions the set of all such pairs into equivalence classes. A rational number

- #5

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HallsofIvy said:nottrue, yes, it is sufficient to give a "counter-example": one case showing for which the statement is not true, thus proving it is notalwaystrue.

What you need to do is find an example of rational numbers (a,b),(a',b'),(c,d),(c',d') such that (a,b) and (a', b') are equivalent, (c,d) and (c',d') are equivalent, but (a+c,b+d) isnotequivalent to (a'+c', b'+d').

(For those who are not clear on this, this is using a method of defining rational number from the integers by saying that two ordered pairs of integers, (a,b) and (a',b') (with second integer non-zero) are equivalent if and only if ab'= a'b. That is an equivalence relation and so partitions the set of all such pairs into equivalence classes. A rational numberissuch an equivalence class.)

That's just what I did, I wasn't just sure if that was enough..

- #6

HallsofIvy

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Out of curiosity then, what was your counter-example?

- #7

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(1,2) ~ (1,2) and (1,3) ~ (2,6).HallsofIvy said:Out of curiosity then, what was your counter-example?

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