if x is any rational number then x^-m =?
Answers
Strictly speaking the statement has undefined meaning (because "1/x is rational" has undefined meaning when x can be 0) rather than that it is false. To make the statement meaningful, you need to ensure it is not talking about the undefined quantity 1/0, and depending on how you do that this can make the statement false or true.
Since in mathematics we are more interested in truths than in falsehoods, when we assert a statement, one usually takes it to also implicitly affirm that everything it talks about is well defined. Applying this to your case would transform the statement into
(For any real number x,) if x is a rational number then 1/x is defined and it is a rational number.
which statement is false (it fails for x=0).
However an other way to repair the statement is to recognise from the outset that x=0 is going to cause an undefined expression to turn up in the formulation, and therefore exclude it, This leads to
For any nonzero real number x, if x is a rational number then 1/x is a rational number.
which statement is true (since Q is a sub-field of R).
The example shows one must always be careful in handling statements that contain potentially undefined expressions.
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