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Exercise 15

Avg difficulty 2.5

Suppose $f:X \to Y$ is a function. Which of the following are possible? Explain.

$f$ is injective but not surjective.
$f$ is surjective but not injective.
$|X| = |Y|$ and $f$ is injective but not surjective.
$|X| = |Y|$ and $f$ is surjective but not injective.
$|X| = |Y|$ , $X$ and $Y$ are finite, and $f$ is injective but not surjective.
$|X| = |Y|$ , $X$ and $Y$ are finite, and $f$ is surjective but not injective.

Hammond, Discrete Math for Shockers, §1.3 Relations and Functions, Exercise 15

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Adapted from Discrete Math for Shockers © 2021 John Hammond. Licensed under CC BY-SA 4.0. Source: Discrete Math for Shockers. Imported at revision 2538beb.