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welcome to this presentation we have already completed our subtractors and different types of errors 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 u X MUX so don't confuse yourself with these two things they are the same so what 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 a 0 i1 i2 i3 by moving this dialer ok 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 this input lines are having the data the information of some activity and by moving this dialer we can have a particular line carrying a particular information selected has the output Y so why not we call it simply a data selector because it is selecting our data ok now there must be one cushion 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 select line or we also call it a selector variable okay in this case when we are having the four inputs there are two selector variables required that is your s 0 and has one as 0 is the LSB of the selector variable and s 1 is the MSB of the selector variable so when the value of s 0 and s 1 are 0 0 let's say this is your s 0 this is your s 1 then the input a 0 will be selected an output Y will be i0 very simple similarly when s 1 and s 0 R 1 1 then put I 3 is being selected as the output Y ok so I think it is an a little bit clearer to you we will see this thing in 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 inputs are mentioned let's say this is I 0 this is I 1 2 inputs are there and remember in multiplexer the number of output is always 1 whatever the number of input be the number of output will always be equal to 1 this multiplexer in which there are two inputs and one output we call it 2 is to 1 MUX okay and there will be a single selector variable as 0 okay sometimes it is also represented like this okay the two inputs one output and one selector variable to cross 1 MUX 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 raised 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 two n so this is the relation between the number of inputs and this selector variable so if n is the number of input then M is this lecture variable in this case the n is equal to four so let's try to find out the value of M it is long to four can be written as 2 to the power 2 and M is equal to 2 long to this thing is equal to 1 so 2 multiplied by 1 gives you 2 so the number of selector lines or select variable is 2 and as you can see that I have already taken as 0 and s 1/2 selector variables in this particular case so this is a little bit of information about the multiplexers 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 ICS okay there are different ICS 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 a full adder or subtractor or any other combinational circuit you can just simply implement it by using a MUX a single IC you don't require much gates in that so there may be great advantages of doing this thing so let's see what are the advantages of using the multiplexer the first advantage that I have written here 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 you 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 ten gates you require so many wires and if you don't have to connect ten gates you have just have to connect one or two gates then you are going to require lesser number of wires so this thing is very true in case of multiplexers 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 caused 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 MUX that I have already explained you and it is most important thing of the multiplexers and when you write your exam definitely one question from the MUX tree that is implementation of different marks by using different marks is there and also the implementation of given logical expression by using the multiplexer is very important in your exams so this is all about the advantages of the multiplexers 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 2 raise to 1 MUX 4 raise to 1 8 raise to 1 16 raise to 1 32 raise to 1 MUX you can see that the output is always 1 this 2 represents the number of input lines and this 1 represents the number of output line so the output is always 1 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 variables in different types of MUX in 2 raise to 1 MUX we are having the sector variable equal to 1 in 4 raise to 1 it is equal to 2 in 8 raised to 1 it is equal to 3 in 16 raised to 1 it is equal to 4 and similarly in 13 raised to 1 it is equal to 5 okay now the next thing and the last thing in this presentation is 2 raise to 1 MUX let's try to study something more about 2 raise to 1 MUX 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 to raise to 1 MUX and the number of selector variable will be single so let's say yes now one more thing is there in this multiplexer is we call II which is your enable now let's see the truth table for the 2 cross 1 MUX 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 walk 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 one and s is equal to zero davai davai is connected to eyes ear oh so Y will be i0 and when the selector variable is equal to one this Y is connected to I 1 and it is equal to i1 simple now let's try to find out the logical expression for this why is equal to what we will see for this two cases and this we are having E and is 1 so e s is 0 so s complement and then I 0 and a 0 or is 1 so e s is also 1 so s and then I 1 you can see that he can be taken as common so let's take ye s common and we are left with s bar and I naught s and I won okay and it is very clear that when the value of e is 0 and it is the end operation the Y is always going to be 0 that I already explained you and the remaining thing is very easy to understand we just write it down from the truth table that we obtained so the next thing is the implementation of this circuit you can easily implement it by using the two and gates these are our two and gates and the N input to this first and gate is what as compliment ize hero in the second and gate the input is simply s and high one and E is input to both of this and gate so E is there then we are having the or 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 to cross one multiplexer