Algebraic Structures | Well-Defined Function
Function Definition:
where ℝ⁺ = positive real numbers
Rule: f(r) = first digit after decimal point in decimal expansion of r
Solution
Definition of Well-Defined Function
A function is well-defined if:
- Every input in the domain produces exactly one output
- If an input has multiple representations, all representations give the same output
For real numbers, this means the output must not depend on which decimal representation we choose.
The function f takes a positive real number r and returns the first digit after the decimal point in its decimal expansion.
Example:
If r = 3.14159…, then f(r) = 1
If r = 0.5, then we need to check: 0.5 = 0.5000… gives f(r) = 5
The problem: Some real numbers have two different valid decimal expansions.
Certain numbers have two decimal expansions:
Terminating decimals have two representations:
Example 1: 0.5 can be written as:
- 0.500000… (infinite zeros)
- 0.499999… (infinite nines)
Example 2: 1.0 can be written as:
- 1.000000… (infinite zeros)
- 0.999999… (infinite nines)
Both expansions represent the same real number.
Take r = 0.5:
Representation 1: 0.5000…
Representation 2: 0.4999…
For any terminating decimal like:
Standard form: N.d₁d₂…dₖ where dₖ ≠ 0
This can also be written as: N.d₁d₂…(dₖ−1)999…
Example: 2.37 = 2.37000… = 2.36999…
- 2.37000… → f = 3
- 2.36999… → f = 3 (still, in this case)
But when borrowing occurs:
1.0 = 1.000… = 0.999…
- 1.000… → f = 0
- 0.999… → f = 9
The existence of even one counterexample (1.0 or 0.5) is enough to show f is not well-defined.
Conclusion
The function f is NOT WELL-DEFINED because:
- Some positive real numbers have multiple decimal expansions
- Different expansions can give different first decimal digits
- The output depends on the representation, not just the number value
- Examples like 0.5 and 1.0 provide clear counterexamples
A well-defined function must give the same output for the same input regardless of how it’s represented.