If $x$ is a Boolean variable then the disjunctions. Different Syntax | Same Semantics for a given interpretation $I$ | Same Boolean functions :------------- |:------------- |:------------- $x\neq$ |$I\models x\Leftrightarrow$ |$f_x=$ $x\vee x\neq$ |$I\models x\vee x\Leftrightarrow$ |$f_{x\vee x}=$ $x\vee x\vee x\neq$ |$I\models x\vee x\vee x\Leftrightarrow$ |$f_{x\vee x\vee x}=$ $x\vee x\vee x\vee x\neq$ |$I\models x\vee x\vee x\vee x\Leftrightarrow$ |$f_{x\vee x\vee x\vee x}=$ $\ldots$|$\ldots$|$\ldots$
are syntactically different Boolean terms (propositions), but they still have the same semantics. Therefore, by definition of Boolean functions, they constitute the same Boolean function.
The same Boolean function might even be represented by propositions containing more than one variable. Let $x$ and $y$ be Boolean variables, Consider the propositions $\phi=x$ and $\psi=(x\wedge y)\vee y$. These propositions have the same truth tables, depending on the semantics of $x$ and $y$:
$[[x]]_I$| $[[y]]_I$| $[[\phi]]_I$| $[[\psi]]_I$ $1$| $1$| $1$| $1$ $0$| $1$| $0$| $0$ $1$| $0$| $1$| $1$ $0$| $0$| $0$| $0$
Therefore, the functions $f_\phi$ and $f_\psi$ are, in fact, the same Boolean function.
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