A common fallacy is mixing up the inclusive disjunction "$\vee$" and the exclusive disjunction "$\oplus$". For two propositions $p$ and $q$, this fallacy takes the following form:
$$\begin{array}{rll} p\vee q&\text{major premise}&\text{e.g. I'll listen to music or I'll learn maths.}\\ p&\text{minor premise}&\text{e.g. I will listen to music.}\\ \hline \neg q&\text{conclusion}&\text{e.g. Therefore, I won't learn maths.}\\ \end{array} $$
Note: This fallacy is quite common since the conjunction "or" in natural language is often used as a "logical exclusive or" rather than "logical inclusive or", which allows for both possibilities to be true ("I'll listen to music and learn maths at the same time").
Proofs: 1