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If [itex]c[/itex] and [itex]c^\dagger[/itex] are fermion annihilation and creation operators, respectively, we know that [itex]cc^\dagger+c^\dagger c=1[/itex] and [itex]cc=0[/itex] and [itex]c^\dagger c^\dagger=0[/itex]. I can use this to show the following

[tex]

[c^\dagger c,c]=c^\dagger cc-c c^\dagger c=-cc^\dagger c=-c(1-cc^\dagger)=-c

[/tex]

But on the other hand I have

[tex]

[c^\dagger c,c]=c^\dagger[c,c]+[c^\dagger,c]c=[c^\dagger,c]c

[/tex]

Does this not imply that [itex][c^\dagger,c]=-1[/itex] and consequently that BOTH the commutator and anti-commutator of [itex]c[/itex] and [itex]c^\dagger[/itex] is equal to unity?