👉 Properly mathematically, we define a function \( f \) from a set \( A \) to a set \( B \) as a rule that assigns to each element \( x \) in \( A \) exactly one element \( f(x) \) in \( B \). For \( f \) to be valid, it must satisfy two key properties:
1.
Well-defined
: For every \( x \in A \), there exists a unique \( f(x) \in B \). This means each input corresponds to a single output, avoiding ambiguity.
2.
Completeness
: The domain \( A \) must be non-empty, ensuring there's at least one element to map. Additionally, \( f \) must map every element in \( A \) to some element in \( B \), covering the entire set \( A \).
These properties ensure a consistent and reliable mapping, crucial for mathematical operations and models.
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