 design circuit as simple as possible involving three switches whose closure properties are given by S1 in the table given. This is the table given in which we have different values for x, y and z which are the three switches and we are given the values for S1 also. Here we have to design a circuit as simple as possible. Let's proceed with the solution now. First of all we need to form a function f1. For this we will consider the values of f1 as 1 only. Now in this case when the value of f1 is 1, x and y are 0 and z is 1. So we can have f1 equal to x complement as x is 0 into y complement as y is 0 into z as z is 1 plus now consider this row in which we have the value of f1 as 1. In this case x is 0, y is 1 and z is 0. So here we have x complement y into z complement plus consider this row where f1 has value 1, x has value 0, y has value 1 and z has value 1. So it would be represented by x complement into y into z plus consider this row in which f1 is 1, x is 1, y is 1 and z is 0. So it would be x into y into z complement plus consider this last row in which we have f1 1, x, y and z are also 1. So it would be represented by x, y, z. So we have formed the function f1 as this. Now we will simplify this function. For this Boolean algebra the sum operation is commutative. So we have a plus d is equal to b plus a this is the commutative law. Now using this commutative law here we get this is equal to x complement y complement z plus x complement y z plus x complement y z complement plus x y z complement plus x y z. Now we have this distributive law that is a into b plus c is equal to a into b plus a into c. Now we have a commutative law that is the operation product is commutative. So a into b is equal to b into a. Using this law in these two terms we get this is equal to x complement z into y complement plus x complement z into y plus x complement y z complement plus x y z complement plus x y z. Now we will apply the distributive law in these two terms. So applying the distributive law now we get this is equal to x complement z into y complement plus y the whole plus x complement y z complement plus x y into z complement plus z the whole. That is we have applied the distributive law for these two terms also. Now further for element a there exists its inverse a complement such that a plus a complement is equal to 1 which is said as a complement plus a and this one is the identity element for the operation of product. So this is further equal to x complement z into 1 plus x complement y z complement plus x y into 1 that is y complement plus y is 1 z complement plus z is 1. Now further a into 1 is equal to a which is equal to 1 into a where this one is the identity element for the operation of the product. So this is equal to x complement z plus x complement y z complement plus x y. Now here also we can apply the distributive law and so this would be equal to x complement into z plus y z complement the whole plus x y. We have another distributive law a plus b into c the whole is equal to a plus c the whole into a plus c the whole. Now we will apply this distributive law for this and so this would be equal to x complement into z plus y the whole into z plus z complement the whole plus x into y. Now further this is equal to x complement into z plus y the whole into 1 that is a plus a complement is 1 so z complement would be 1 this whole plus x into y. Now further we have x complement into z plus y the whole plus x into y. Now where plus y the whole into 1 would be equal to z plus y. Now we will apply the distributive law here so this is equal to x complement into z plus x complement into y plus x into y. Further x complement into z plus again applying the distributive law here would give us x complement plus x the whole into y. So this is equal to x complement into z plus 1 into y that is x complement plus x is 1 so further we have x complement into z plus y this 1 into y would be y. So this is the function f1. This is the simplified form of the function f1. Now we will design a circuit for this. This is the circuit for the function f1 that is x complement into z which means that the switches x complement and z are in series plus y shows that y is in parallel with the network of x complement and z. So this completes the session. Hope you have understood the solution of this question.