Mathematical Writing - Vivaldi Franco 2014
Wrong Definitions
Existence and Definitions
As we did in Sect. 7.7 for logical arguments, we consider here some common mistakes made in definitions. A faulty definition may imply that the object being defined does not exist at all, or that it is not the one we had in mind, or that there is more than one object that fits the description.
We begin with an incorrect function definition which can be put right in several ways.
WRONG DEFINITION Let be given by:
The mistake is plain but fatal: the expression involves the unspecified quantity . As defined, is not a function. There are many legitimate interpretations of what this formula could mean, each resulting in a different function.
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In the second example, the existence of a function requires more conditions than those given. The definition relies on a hidden assumption, so that using it adds extra information (we say that the definition is creative).
WRONG DEFINITION Let and be sets, and let be functions. We define the function as follows:
The hidden assumption is that the elements of the co-domain can be added together. However, may be a set where addition is not defined (e.g., and are predicates) or which is not closed under addition (e.g., is an interval). We give a more restrictive definition without hidden assumptions.
DEFINITION Let be a set, let be a set closed under addition, and let be functions. ...
In the next example we attempt to extend to modular arithmetic the concept of the reciprocal of an element, but our definition has a hidden assumption.
WRONG DEFINITION Let be a natural number, and let be an integer. If , then we define the multiplicative inverse as follows:
Take and . We have , but the equation has no solution. What went wrong? We have assumed that the implication
which is valid for real or complex numbers, is also valid for congruence classes. Requiring that , namely that is not divisible by the modulus is not enough; for this implication to hold, we must assume that is co-prime to the modulus.
DEFINITION Let be a natural number and let be an integer co-prime to . We define the multiplicative inverse of as follows...
In the following definition both existence and uniqueness are problematic.
WRONG DEFINITION Let , and let be the root of having smallest modulus.
If has degree zero, then does not exist, so this definition relies on the hidden assumption that has positive degree. If exists, then it may still not be unique (e.g., ), but the use of the definite article (’the root’) implies that it is. Requiring that be non-constant solves the existence problem. The way we address uniqueness depends on the context. If we merely require that no root of has smaller modulus than , then we don’t need uniqueness.
DEFINITION Let with , and let be a root of with smallest modulus.
This definition does not identify ; it tells us that is a member of a well-defined non-empty set, but this set may have more than one element. If, on the other hand, we require a specific complex number, then we must supply additional information. In the following definition we impose a condition on the argument of , which removes any ambiguity.
DEFINITION Let be a non-constant polynomial with integer coefficients, and let be the root of with smallest modulus. If there is more than one root with this property, we let be the root with smallest non-negative argument.
In our final example, the definition hides a subtle lack of uniqueness.
WRONG DEFINITION Let , and let be the function that associates to its binary digits sequence
where
(9.10)
(The co-domain of is the set of all infinite binary sequences, see Sect. 3.5.1.) What’s wrong with this definition? Consider the rational numbers
If we write in the form (9.10), then the above identity shows that there are two binary representations, namely
(9.11)
So the function is not uniquely defined at these rationals and, more generally, at all rationals of the form
where are arbitrary binary digits.
We resolve this ambiguity by choosing consistently the first of the two digit sequences in (9.11).
DEFINITION Let be given by
Without loss of generality, we assume that all sequences contain infinitely many zeros. Let now f be the function that associates to its binary digit sequence
Exercise 9.1
Use Dirichlet’s box principle to prove the following existence statements.
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Exercise 9.2
Each function definition contains an error. Explain what it is and how it should be corrected.
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Exercise 9.3
The following definitions have several flaws. Explain what they are, hence write a correct, clearer definition.
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Exercise 9.4
Exploit each conjecture to define a function whose existence cannot be decided at present.
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Footnotes
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There was no guess here: first I chose , and then I derived the equation of which is a solution.
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Some authors use ’effective’ to mean ’constructive’.
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For , the integer is followed by composite integers. (Why?)
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Conjectured by Joseph Bertrand (French: 1822—1900) in 1845. Proved by Pafnuty Chebyshev (Russian: 1821—1894) in 1850.
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Leonardo Pisano, known as Fibonacci (Italian: 1170—ca.1240).