iff. Prove this is an equivalence relation. Reflexive: ; however, does ? I was under the impression
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Originally Posted by dwsmith iff. Prove this is an equivalence relation. Reflexive: ; however, does ? I was under the impression the ratio of two integers.
Originally Posted by Plato the ratio of two integers. I understand that but I thought were all such that
Symmetric: , then Since , then a and b can expressed as and How can I get than in the form of ? Would it be allowable to multiple through by a -1 and then swap cd and ef to obtain: ?
Originally Posted by dwsmith I understand that but I thought were all such that It is .
Originally Posted by dwsmith Symmetric: , then Since , then a and b can expressed as and How can I get than in the form of ? Would it be allowable to multiple through by a -1 and then swap cd and ef to obtain: ? Transitive: , then add together Equivalence class of and a and Correct?
Last edited by dwsmith; Apr 29th 2010 at 04:45 PM.
The sum of two rationals is rational. So it is transitive. Way to go!
What about what I did for symmetric? Is that legal?
Originally Posted by dwsmith What about what I did for symmetric? Is that legal? Of good grief, come on THINK. If then
Originally Posted by dwsmith Symmetric: , then Since , then a and b can expressed as and How can I get than in the form of ? Would it be allowable to multiple through by a -1 and then swap cd and ef to obtain: ? If and it means, by definition, that such that . What can you say about then? Your proofs for symmetry and transitivity are both incorrect -- means that , but that doesn't necessarily mean that -- take, for example, ...
Originally Posted by Defunkt If and it means, by definition, that such that . What can you say about then? Your proofs for symmetry and transitivity are both incorrect -- means that , but that doesn't necessarily mean that -- take, for example, ... I understand what you mean; however, I am not sure on how to show it if what I have done is incorrect.
For symmetry you only have to notice what Plato said - if then . For transitivity, again, as Plato said -- the sum of two rationals is rational. How can you express in terms of two numbers you know are rational?
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