When is the empty function injective? surjective? bijective?
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\(\bullet\) Show that \(\forall X,f:\emptyset\to X\) is injective
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\(\Vdash\) \(\forall X,f:\emptyset\to X\) is injective
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\(\equiv\) { Definition 3.3.14 }
- \(\forall X, \forall x\in\emptyset, \forall x'\in\emptyset,~x\neq x'\implies f(x)\neq f(x')\)
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\(\equiv\) { \(x,x'\in\emptyset\) is always false }
- Vacuosly true
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\(\square\)
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\(\bullet\) Show that \(f:\emptyset\to \emptyset\) is surjective
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\(\Vdash\) \(f:\emptyset\to\emptyset\) is surjective
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\(\equiv\) { Definition 3.3.17 }
- \(\forall y\in \emptyset, \exists x\in\emptyset,~f(x) = y\)
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\(\equiv\) { \(\forall y\in\emptyset\) is always false }
- Vacuosly true
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\(\square\)
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\(f:\emptyset\to\emptyset\) is bijective by Definition 3.3.20 \(\square\)