Understanding “Specified on Elements” Functions
When a function is defined by explicitly stating what happens to each individual element, we call it “specified on elements.”
Mathematical Examples:
1. \( f: \mathbb{Z} \to \mathbb{Z} \) defined by \( f(n) = 2n \)
2. \( \varphi: S_3 \to \{1, -1\} \) defined by \( \varphi(\sigma) = \text{sign}(\sigma) \)
3. Inclusion map \( i: H \to G \) defined by \( i(h) = h \)
Important: For functions specified on elements, well-definedness is automatic because each input gets exactly one clearly defined output. Nobody asks “but is it well-defined?” in these cases.
Understanding “Not Specified on Elements” Functions
These functions are defined by rules or using representatives rather than explicit element-by-element specification.
Critical Example – The Overlap Problem:
Let \( A = A_1 \cup A_2 \) and define \( f: A \to \{0, 1\} \) by:
• If \( a \in A_1 \) then \( f(a) = 0 \)
• If \( a \in A_2 \) then \( f(a) = 1 \)
Problem: If \( A_1 \cap A_2 \neq \emptyset \), elements in the intersection get two different outputs! This proposed definition only gives a proper function if \( A_1 \cap A_2 = \emptyset \).
Abstract Algebra Examples
Classic Cases Requiring Verification:
Quotient Group Multiplication: \( (aH) \cdot (bH) = (ab)H \)
Must check: If \( aH = a’H \) and \( bH = b’H \), then \( (ab)H = (a’b’)H \)
This definition can be made well-defined if and only if \( H \) is a normal subgroup.
Modular Arithmetic Map: \( f([k]) = k \mod n \)
Must check: If \( [k] = [l] \), then \( k \mod n = l \mod n \)
Field of Fractions: \( a/b + c/d = (ad + bc)/(bd) \)
Must check: Different representatives give same result
Verification Examples
Example 1: Modular Arithmetic ✅
Define \( f([x]) = x^2 \mod 5 \)
Check: If \( [7] = [2] \) (since 7 ≡ 2 mod 5), then:
• \( 7^2 \mod 5 = 49 \mod 5 = 4 \)
• \( 2^2 \mod 5 = 4 \mod 5 = 4 \)
✅ Results match → Function is well-defined
Example 2: Piecewise Function ✅
Define \( f(x) = \begin{cases} x & \text{if } x \geq 0 \\ -x & \text{if } x < 0 \end{cases} \)
Check boundary at \( x = 0 \):
• First rule: \( f(0) = 0 \) (since 0 ≥ 0)
• Second rule: Doesn’t apply (0 is not < 0)
✅ No conflict → Function is well-defined
Comparison and Practical Checklist
| Aspect | Specified on Elements | Not Specified on Elements |
|---|---|---|
| Definition | Explicit \( f(a) = b \) for each \( a \) | Defined by rules/representatives |
| Ambiguity | None | Possible, must check |
| Well-definedness | Automatic | Must be verified |
| Common in | Concrete functions | Abstract algebra constructions |
Practical Checklist for Verification:
- Take two representations of the same element
- Apply the rule to both representations
- Check if results match
- Consider all edge cases (boundaries, overlaps)
Summary
Key Takeaways:
- ✅ Functions “specified on elements” have automatic well-definedness
- ✅ Functions “not specified on elements” require explicit verification
- ✅ Most abstract algebra constructions (quotient groups, modular arithmetic) fall into the second category
- ✅ Always check for overlaps, boundary cases, and representative independence
Easy Memory Aid:
“Specified on elements” = Roll number-wise marks sheet (each student’s marks listed separately)
“Not specified on elements” = Grading system with boundary cases (“90% and above: A”, “80-90%: B”)