 Hello and welcome to the session. In this session, first we discuss the OR switching circuit or P plus Q circuit. We consider two switches P and Q in parallel. As you can see in this circuit, the switches P and Q are in parallel. This B is the battery, L is the lamp and this network is called a basic parallel network. Now, there are two raise to be power two that is four possible states this circuit in which the two switches P and Q are in parallel. In this first case, as you can see in this circuit both the switches P and Q are open. In the second state, as you can see the switch P is open, the switch Q is closed. In this case, the switch P is closed and the switch Q is open. Now in this case, both the switches P and Q are closed. Now in the first case, both the switches P and Q are open. So P would be zero, Q would be zero and as both the switches are open, so no current will flow and as no current will flow, so the lamp will not light and so here we have zero. In this second case, switch P is open and switch Q is closed. Now as P is open, so here we have zero, Q is closed, so here we have one. Now as P and Q are parallel and one of the two switches is closed, so the current would flow and as the current flows the lamp would light and so here we have one. Now in the third case, we have switch P is closed and switch Q is open. So P will have the value one, Q will have the value zero. As one of the two switches is closed, so the current would flow and so the lamp would light and so value of L here would be one. In this fourth case, both the switches P and Q are closed, so the values of P and Q would be one and one and so the current would flow and the lamp would light and so value of L here would be one. So this circuit in which we have the switches P and Q in parallel is called as the P plus Q circuit and OR circuit because from this table we observe that the lamp lights if either P or Q or both are closed. If in case we have three switches in parallel, then in that case there would be two rays to the power three as there are three switches which is eight possibilities and those eight possibilities could also be shown in the same way as we have shown for the two switches in parallel. Next we discuss complex circuit. In complex circuits, some switches are coupled together. There are two coupling. The one in which the switches are either all open or closed. In this case, the switches are denoted by same symbol. There is another coupling. There is another coupling in which some switches are open, others closed and vice versa and in this case one set is denoted by its complement. The switches not included in coupling are denoted by different symbols. This figure shows the coupling of switches. Now these switches P which are denoted by the same symbol P will either be both open or both closed. As we know that when the switches are denoted by the same symbol it means that they are either all open or closed and when some switches are open and others closed and vice versa then in that case we denote one set by its complement as in this figure we have a switch Q and its complement Q complement. So this means when switch Q is open then Q complement is closed vice versa and we have the switch R which is not included in the coupling and so it is denoted by a different symbol R. Now the circuits in which the switches in parallel are combined in the same circuit switching circuits. It means a network which contains the AND circuits as well as the OR circuits. Consider the circuit which contains both the AND and OR circuits. Here the switches P complement and Q are parallel and it is in series with the switch P. Let us now try to simplify the circuit. So let us do the circuit simplification. We have made this table to show the states of the switches P, Q, P complement and the lamp L. Now when the switch P is open and switch Q is open then switch P complement would be closed as switch P is open and so from the circuit we observe that the lamp L would not light up in this case and so it would have the value 0. Now when the switch P is open switch Q is closed then switch P complement would be closed and in this case the lamp L would not light up. Next is the case when switch P is closed switch Q is open and switch complement in this case would be open and the lamp L would not light up again. Now the case when switch P is closed switch Q is closed and switch P complement would be open. In this case the lamp L would light up. Now from this table we find that the lamp L lights up only when both the switches and Q and so the given circuit would be equivalent to this circuit in which the switches P and Q are in series and they both are closed in this case the lamp L would light up. So this is one way of simplifying a given circuit. We can also simplify the given circuit using Boolean algebra. Now this given circuit can be written in Boolean algebra P into P complement plus Q the whole. Now P complement and Q are parallel so there is a plus sign between them and it is in series with the switch P so there is a multiplication sign between them. So this is the given circuit now this is equal to P into P complement plus P into Q. Now P into P complement would be 0 plus PQ this is by the property of inverse that P into P complement is 0. Now 0 plus PQ would be equal to PQ this is by the property of identity. So we have the given circuit P into P complement plus Q the whole is equal to PQ and this circuit shows the expression PQ in Boolean algebra. Let us now discuss design of circuits. We have studied the similarities of switching circuits with Boolean algebra. Thus the laws of the two element Boolean algebra can be used to produce and modify electrical circuits designed to behave in a particular way. Suppose we need to design a circuit which operates both the switches are on or when both the switches but does not operate when one switch is on and the other is off. This condition to design a circuit shows that we have two switches so we suppose x and y to be the two switches in the circuit. Now we will make a truth table and summarize these operations. Now when switch x is off that is it has value 0 and switch y is also off that is it also has the value 0. Now as both the switches are off so it means that the circuit will operate and so f of x, y would have the value 1. Now when switch x is on that is it has value 1 and switch y is off that is it has value 0. Then in this case according to the condition given we have that the circuit will not operate when one switch is on and the other switch is off. So it means the value of f of x, y would be 0. Now x consider when switch x is off and switch y is on and in this case also the circuit will not operate and so f of x, y would be 0. Now when both the switches x and y are on that is x has value 1, y has value 1. Then in that case the circuit will operate and so f of x, y would be 1. Now this table shows that f of x, y is equal to x complement y complement plus y. This shows that x complement y complement are in series that is they are in one line. And since there is a plus sign between these two term so this x, y would be in parallel with x complement y complement. So this is the complete circuit which is a parallel circuit as you can see x complement y complement are in series, x and y are also in series and they both are parallel to each other. This is how we can design a circuit. Next we discuss closure tables. Two Boolean functions can be proved equal using closure tables. Let us now try to prove p plus q into r the whole equal to p plus q the whole into p plus r the whole. So these are the two Boolean functions that we need to prove equal using the closure tables. Now from this we observe that there are three switches p, q and r there would be two raised to the power three that is eight possibilities. Consider this table. Now we would use these laws of Boolean algebra to complete this table. If we take p0, q0, r0 then q into r would be 0 into 0 which would be equal to 0 using this law. Then p plus q into r the whole that is 0 plus 0 would be again 0 using this law. Then p plus q which is 0 plus 0 would be again 0 using this law. Then p plus r which is 0 plus 0 would be again 0. Then p plus q the whole which is 0 into p plus r the whole that is 0. So 0 into 0 is also 0. Now we take p0, q0 and r1 then q into r that is 0 into 1 would be 0 using this law that is 1 into 0 is same as 0 into 1. By commutativity of product then p plus q into r the whole that is 0 plus 0 would be 0. In the same way p plus q would be 0, p plus r would be 1. Now p plus q that is 0 into p plus r that is 1 would be 0 using this law. In the same way we fill the table when p is 0, q is 1 and r is 0. Next we fill for p0, q1, r1. In the same way we fill the rest of the table using these laws of Boolean algebra. As you can observe from the table that this column of p plus q into r the whole and this column of p plus q the whole into p plus r the whole are identical. Therefore the two functions p plus q into r the whole and p plus q the whole into q plus r the whole are equal. So we have proved the two Boolean functions to be equal using the closure tables. In this complete C session hope you have understood the OR circuit, complex circuit, design of circuits and the closure tables.