 Hello and welcome to the session. In this session, we discuss the following question which says x, y, z represent the three switches in an on position construct a switching circuit representing the polynomial x into y plus z the whole into y plus z the whole plus z. Use Boolean algebra to construct an equivalent simplified circuit. Let's proceed with the solution now. We are given the polynomial x into y plus z the whole into y plus z the whole plus z. Where these x, y and z are three switches which are in the on position and we are supposed to now construct a switching circuit which would represent this polynomial. Now as you can see that there is a plus sign between these two expressions so these two networks would be parallel and as these two expressions are multiplied with each other so they would be in series. Now consider the expression x into y plus z x into y means that the switches x and y would be in series and this network would be parallel to the switch z. So here we have the switches x and y in series and this network is parallel to the switch z. Now consider this y plus z this shows that y and z switches would be parallel to each other and they would be in series with this network. So here we have the switches y and z which are parallel to each other but in series with this network. Here we have a switch z so this is the switching circuit which represents the polynomial x into y plus z the whole into y plus z this whole plus z. Now consider the polynomial x into y plus z the whole into y plus z the whole plus z. Now we will simplify this polynomial. Now we have a distributive law according to which we have a plus b into c the whole is equal to a plus b the whole into a plus c the whole. For applying this distributive law for this expression we get this is equal to z plus x into y into y the whole and this whole plus z. Now from the idempotent law we have a into a is equal to a so this is equal to z plus x into y that is y into y is y so here we have x into y this whole plus z. Now the sum operation is commutative that is a plus b is equal to b plus a this is the commutative law so plus x into y could be written as x into y plus z this whole plus z. Now further we have x into y plus z plus z. We have another idempotent law according to which we have a plus a is equal to a further we can write this as x into y plus z that is z plus z is written as z. So now the given polynomial x into y plus z the whole into y plus z the whole plus z equal into x into y plus z. Now we will draw the simplified circuit. Now x into y plus z means that there are two networks parallel to each other one is the network containing the switches x and y the other is the network containing the switch z. Now x into y means that the switches x and y are in series so this is the simplified circuit which represents x into y plus z. This completes the session hope we have understood the solution of this