Answer : Half reaction is defined as the reaction which will be either the oxidation or reduction reaction of particular species in a given redox reaction.
So as per the definition in the reaction of,
[tex]Ba + ZnSO_{4} ----\ \textgreater \ BaSO_{4} + Zn[/tex]
[tex]Ba_{(s)} + Zn ^{+2} + SO_{4}^{-2} = Ba^{+2} + SO_{4}^{-2} + Zn_{(s)}[/tex]
here on left and right side [tex]SO_{4}^{-2}[/tex] will get cancelled.
and we will get, [tex]Ba_{(s)} + Zn ^{+2} = Ba^{+2} + Zn_{(s)}[/tex]
Here,
[tex]Ba_{(s)} ----\ \textgreater \ Ba^{+2} - 2e^{-}[/tex] ...this is oxidation reaction as Ba is getting reduced to [tex]Ba^{+2}[/tex] by losing 2 electrons.
whereas,on the other hand, [tex]Zn ^{+2} ----\ \textgreater \ Zn_{(s)} + 2e ^{-} [/tex]...this is reduction reaction as [tex]Zn ^{+2}[/tex] is getting reduced to Zn by gaining 2 electrons.