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Function Restriction: Uniqueness of Images
publish date: 2026/05/23 22:20:50.505983 UTC
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Why is \(R = \{(1, 2), (1, 3), (2, 5)\}\) not a function from \(\{1, 2\}\) to \(\{2, 3, 5\}\)?
Correct Answer
The element 1 in the domain has two different images (2 and 3)
Explanation
For a relation to be a function, each domain element must be associated with exactly one element in the codomain. Here, \(1 \in A\) appears in two ordered pairs with different second elements: \((1, 2)\) and \((1, 3)\). This violates the uniqueness condition. It does not matter that 5 has no pre-image — surjectivity is not required for a general function.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012