Simplify the following circuit so that new circuit has minimum number of switches. Also draw simplified circuit.

Explanation

Let p : The switch `S_(1)` is closed
q : The switch `S_(2)` is closed
`~p`: The switch `S_(1)` is open or the switch `S_(1)` is closed
`~q` : The switch `S_(2)` is open or the switch `S_(2)` is closed
`:.` The given circuit in the symbolic form is
`(p^^ ~ q) vv (~p ^^ q) vv (~ p ^^ ~q)`
`-=(p^^~q)vv [(~p^^q)vv(~p^^~q)]` [By Associative law]
`-=(p^^ ~ q)vv[~p^^(qvv~q)]` [By Distributive law]
`-=(p ^^ ~ q) vv (~p ^^ T)` [By Complement law ]
`-= ( p ^^ ~ q) vv (~p)` [ By Identity law]
`-+ (p vv ~ q ) ^^ (~ q vv ~ p)` [By Distributive law ]
`-= T ^^ (~ q vv ~ p)` [ By Complement law]
`-=~q vv ~ p` [ By Identity law]
`-= ~ p vv ~ q` [By Commutative law ]
Hence, the alternative arrangement for the given circuit is .