When brine solution is subjected to electrolysis the gases liberated at anode and cathode are

$2NaCl\stackrel{electrolysis}{\to }2N{a}^{+}+2C{l}^{-}$
At anode :  $2C{l}^{-}\to C{l}_2\uparrow +2e^{-}$
At cathode :  $2H_{2}O\to 2H^{+}+2O{H}^{-}$
$2H^{+}+2e^{-}\to H_2\uparrow$
$2N{a}^{+}+2O{H}^{-}\to 2NaOH$