# Co-NP definition

The class Co-NP is defined as all the languages $$L$$ such that $$\overline{L} \in NP$$. An example that appears in the book of Arora and Barak is of $$\overline{SAT}$$, which is defined as $$\overline{SAT} = \{\phi : \phi \text{ is not satisfiable} \}$$. My question is the following, given that $$SAT$$ corresponds to boolean functions represented with an alphabet (say $$\{0,1\}$$), then $$\overline{SAT}$$ should include bitstrings that correspond to non-satisfiable formulas but also butstrings that dont represent a valid boolean formula. Why is then in the book defiend as $$\overline{SAT} = \{\phi : \phi \text{ is not satisfiable} \}$$?

Shouldn't the definition be something like $$\overline{SAT} = \{\phi : \phi \text{ is not satisfiable} \} \cup \{\phi : \phi \text{ does not encode a boolean formula} \}$$.