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Empty Relation as a Function
publish date: 2026/05/23 22:20:51.396003 UTC
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Let \(A = \{1, 2\}\) and \(B = \{3, 4\}\). Is the empty relation \(R = \varnothing\) a function from \(A\) to \(B\)?
Correct Answer
No, because it violates the condition that every element of \(A\) must have an image
Explanation
A function from \(A\) to \(B\) requires that every element of \(A\) appears as a first component in exactly one ordered pair. The empty relation contains no ordered pairs, so elements 1 and 2 have no image. Therefore, it is not a function from \(A\) to \(B\). However, if \(A = \varnothing\), then the empty relation is a function (the empty function) because the condition "every element of A has an image" is vacuously true.
Reference
Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012