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## HypothesesHypotheses and theorems related to functions.## Sibling topics:## Contents:- Domain/codomain cardinality given a bijective function
- Empty codomain implies empty domain
- Empty domain implies empty function and image set
- Number of distinct bijective functions for a finite domain and codomain
- Number of distinct functions for a finite domain and codomain
- Number of distinct injective functions for a finite domain and codomain
- Number of distinct surjective functions for a finite domain and codomain
- Surjectivity does not imply injectivity and vice versa in infinite, equal domains and codomains
- Surjectivity implies injectivity and vice versa for finite, equal domains and codomains
Hypothesis: Domain/codomain cardinality given a bijective function Hypothesis: Surjectivity implies injectivity and vice versa for finite, equal domains and codomains Corollary (of
Bijectivity in finite, equal domains/codomains):
Surjectivity does not imply injectivity and vice versa in infinite, equal domains and codomains Theorem: Empty codomain implies empty domain Given `f:A->B`, if `B=O/`, then `A=O/`. Assume by way of contradiction that `B=O/` but `A!=O/`. Then there is an element `u in A`. By the
the definition of
a function, there exists an element `v in B` such that
`(u,v) in A xx B`. But `B=O/` so there can be no such element. Therefore, if `B=O/`, then `A=O/`.Proof: Theorem: Empty domain implies empty function and image set Given `f:A->B`, if `A=O/`, then `f=O/ and Im(f)=O/`. This theorem is really two theorems in one, so we'll address each separately.
Proof: - Assume by way of contradiction that `A=O/` but `f!=O/`. Then by the definition of a function, there is an element `(u,v) in A xx B`, but because `A=O/`, this cannot be the case. Therefore, if `A=O/`, then `B=O/`.
- Assume by way of contradiction that `A=O/` but `Im(f)!=O/`. Then, there exists an element `v` in the image set. By the definition of image set, there exists an element `u in A` such that `f(u)=v`. But since `A=O/`, that cannot be the case. Therefore, if `A=O/`, then `Im(f)=O/`.
Hypothesis: Number of distinct functions for a finite domain and codomain Hypothesis: Number of distinct injective functions for a finite domain and codomain Hypothesis: Number of distinct surjective functions for a finite domain and codomain Hypothesis: Number of distinct bijective functions for a finite domain and codomain |