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I was wondering that since acids dissociate in water to make a lot of $\ce{H+}$ cations. Is it possible that introducing a supply of electrons via an electric current, would create $\ce{H2}$ gas?

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Yes. In fact the first electrolysis was done with acidified water. Pure water is only slightly ionized and is a poor conductor. Adding ions to water increases its conductivity and speeds up the electrolysis process. So by adding a source of protons to pure water the following reaction will occur at the cathode $$\ce{2H+ (aq) + 2e^{-}->H2(g)}$$ Of course a parallel reaction will occur at the anode $$\ce{4OH- ->2H2O + O2(g) + 4e-}$$ otherwise a current wouldn't flow and the electrolysis would not continue.

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