welcome to this presentation we have already completed our subtractors and different types of adders now this is time to move to our next combinational circuit that we call as the multiplexers sometimes you will find multiplexers written as M uux marks so don't confuse yourself with these two things they are the same so what is is a multiplexer it is a combinational circuit that selects binary information from one of many input lines and directs it to the output line here you can see this thing in which there are four inputs and one output and a dialer is there it means you can select any input i0 i1 I2 or I3 by moving this dialer okay at this position the position you can see here the output Y is going to be i0 if I move it here at this point then the Y is going to be I2 so you already know that these input lines are having the data the information of some activity and by moving this dialer we can have a particular line getting a particular information selected as the output y so why not we call it simply a data selector because it is selecting a data okay now there must be one question arising in your mind that how this dialer is moving how out of these four inputs one of the input is being selected so it is done by a select line uh select line or we also called it as selector variable okay in this case when we are having the four inputs there are two selector variables required that is your s0 and S1 s0 is the LSB of the selector variable and S1 is the MSB of the selector variable so when the value of s0 and S1 are 0 0 let's say this is your s0 this is your S1 then the input i0 will be selected and output y will be i0 very simple similarly when S1 and s0 are 1 one the input I3 is being selected as the output y okay so I think it is now a little bit clear to you we will see this thing in U great detail in the coming presentations so the next thing that we have to see is the representation of the multiplexer it is represented by this rectangular box okay in which the number of of inputs are mentioned let's say this is i0 this is i1 two inputs are there and remember in multiplexer the number of output is always one whatever the number of input be the number of output will always be equal to one this multiplexor in which there are two inputs and one output we call it 2 to 1 marks okay and there will be a single selector variable as zero okay sometime it it is also represented like this okay the two inputs one output and one selector variable two cross one marks okay these two things this one and this one are the same representing the multiplexer that is 2 is to 1 now the another important thing is the relation between the selector variable and the inputs what is the relation between the selector variables and the input so let's say if n is the number of input and it is equal to 2 to the power M then M this m will give you the number of Select lines and this n is your number of input okay so if if I simplify it it will be something like m is equal to log with base 2 N so this is the relation between the number of inputs and the selector variable so if n is the number of input then m is the selector variable in this case the N is equal to 4 so let's try to find out the value of M it is log 2 4 can be written as 2 to the^ 2 and M is = to 2 log 2 this thing is equal to 1 so 2 * by 1 gives you two so the number of selector lines or select variable is two and as you can see that I have already taken s0 and S1 two selector variables in this particular case so this is a little bit of information about the multiplexors now you might be thinking that why to select one of the input out of many there are so many so many and so many uses of these multiplexers it is a medium scale integrated circuit and it is available in the form of icc's okay there are different IC for different types of multiplexers available in the market and by using a single IC you can Implement various number of what combinational circuits like you want to implement a half adder U full L A subtractor or any other combinational circuit you can just simply implement it by using a MX a single IC you don't require much gates in that so there may be great advantages of doing this things so let's see what are the advantages of using the multiplexers the First Advantage that I have written here is it reduces the number of wires I just told you that we don't require number of gates you are just going to implement your logic your combinational logic by using a single IC so why you are going to require more wires because you're not going to connect number of gates for example if you have to connect 10 Gates you require so many wires and if you don't have to connect 10 Gates you just have to connect one or two gates then you are going to require lesser number of wires so this thing is uh very true in case of multiplexors the another thing is it reduces the circuit complexity and cost let's see how I just told you that the wires are reduced and wires are reduced because the gates are reduced when both wires and gates are reduced definitely the complexity is going to reduce and at the same time cost because Gates and wires have definitely some cost so this the second point is also true now the third point is implementation of various circuit using the marks that I have already explained you and it is most important thing of the multiplexors and when you write your exam definitely one question from from the M tree that is implementation of different Ms by using different Ms is there and also the implementation of given logical expression by using the multiplexor is very important in your exams so this is all about the advantages of the multiplexer now the next thing that we have to do is the types what are the different types of multiplexers that we have to cover in this particular subject it is two ra to one marks 4 ra to 1 8 ra to 1 16 ra to 1 32 ra to 1 marks you can see that the output is always 1 this two represents the number of input lines and this one represents the number of output line so the output is always one you can see and I have already told you this thing at the beginning of this lecture now let us try to guess what is the number of selector variable in different types of marks in 2 1 marks we are having the selector variable equal to 1 in 4 to 1 it is equal to 2 in 8 to 1 it is equal to 3 in 16 to 1 it is equal to 4 and similarly in 13 to 1 it is equal to 5 okay now the next thing and the last thing in this presentation is 2 to 1 marks let's try to study something more about 2 to 1 marks so as I told you it is represented by a rectangular box in which the number of inputs are shown this is i0 i1 the output is shown Y is the output 2 to 1 Max and uh the number of selector variable will be single so let's say s now one more thing is there in this multi complexer is we call E which is your enable now let's see the truth table for the 2 cross one marks this is your enable this is your s selector variable and this is your output why now when the value of e is equal to zero this circuit will not work and when the circuit is not working the output is equal to zero so whatever the value of selector variable we don't care it is always going to be zero when the enable is zero now when the enable is 1 and S is equal to Z the Y the Y is connected to i0 so y will be i0 and when this elector variable is equal to 1 this Y is connected to i1 and it is is equal to i1 simple now let's try to find out the logical expression for this Y is equal to what we will see for this two cases and this we are having e and e is one so e s is zero so s complement and then I Z and i z or E is 1 so e s s is also one so s and then i1 you can see that e can be taken as common so let's take e as common and we are left with S Bar and I KN s and i1 okay and it is very clear that when the value of e is zero and it is the end operation the Y is always going to be zero that I already explained you and the remaining thing is very easy to understand we just write it down from the truth table that obtained so the next thing is the implementation of this circuit you can easily implement it by using the two end Gates these are our two end Gates and the in input to this first end gate is what s complement i0 in the second end gate the input is simply s and i1 and E is input to both of this and gate so e is there then we are having the r combination of this two and Gates output so this one is your y a very simple thing to do so this is all that we need to study in the 2 cross one multiplexor