 Hello and welcome to the session. In this session we discussed the following question which says, a committee of three deciding on the acceptance or rejection of a project for a dam are provided with buzzers, which they push to indicate acceptance, design a circuit so that a bell will ring when there is a majority vote for acceptance, but the acceptance of the first member is must. Let us now proceed with the solution. First of all, let X, Y and Z be the buzzers which are given to the committee members which they will push to indicate the acceptance. One would represent a vote for acceptance and zero would represent a vote for rejection. And we suppose let F be the required function and now we will draw a closure table for the function F as in the question it's given that a bell will ring when there is a majority vote for acceptance and the acceptance of the first member is must. And we have taken one to denote vote for acceptance, zero to denote vote for rejection. So, this function F is given the value one if at least X, Y and Z have value one and X value one. So in this case F would be given the value one that is when at least two of X, Y or Z have values one and X should necessarily have value one and in all other cases F has value zero. Now, when X is zero, Y is zero and Z is zero that is all the three members show the rejection. So in this case F would be zero that is the bell will not ring. Now, when X is zero, Y is zero and Z is one that is only the third member accepts the project. So in this case also F would be zero as it does not show the majority of acceptance and the first member also shows rejection and not acceptance. Now in the next case when X is zero, Y is one, Z is zero then in this case also F would be zero as only the second member accepts the proposal and all other members show rejection. Then in next case when X is zero, Y is one, Z is one then in this case also F would be zero where two members out of the three members shows acceptance but the first member shows rejection. So in this case F is zero that is the bell will not ring. Then next when X is one, Y is zero, Z is zero that is only the first member is accepting the proposal then in this case also F would be zero that is bell will not ring. Then next when X is one, Y is zero and Z is one that is in this case two members the first member and the third member show acceptance and so as the majority of votes go for acceptance so F would be one in this case and as you can see the first member has a vote for acceptance and so in this case F is one. Now next when X is one, Y is one and Z is zero then in this case also F would be one as there is majority of votes for acceptance and the first member also has a vote for acceptance. Now next when X is one, Y is one, Z is one then in this case F would be one as all the three members give the vote for acceptance and the first member also gives the vote for acceptance. So this is the closure table for the function F. Now from this table we will form the function F and we will consider the case only when F is one so these are the three cases when F is one so using this we form the function F as X into Y complement into Z plus into Y into Z complement plus X into Y into Z. Now let's try to simplify this function F that is consider X into Y complement into Z plus X into Y into Z complement plus X into Y into Z. Now this is same as X into Y into Z plus X into Y complement into Z plus X into Y into Z complement. Now for this Boolean algebra we have this distributive law A into B plus C the whole is equal to A into B plus A into C. You also know that the product operation is commutative that is A into B is equal to B into A this is the commutative law so using this commutative law we have here X into Z into Y plus X into Z into Y complement plus X into Y into Z complement. Now we will apply the distributive law here that is this law so applying this law we get this equal to X into Z into Y plus Y complement the whole plus X into Y into Z complement. Now for an element A which belongs to the set B there exists its inverse A complement such that A plus A complement is equal to 1 and this 1 is the identity element for the operation of product so this would be equal to X into Z into 1 the whole plus X into Y into Z complement this is using A plus A complement is equal to 1 so Y plus Y complement would be equal to 1. Now A into 1 is equal to 1 into A and this would be equal to A where this 1 is the identity element for the operation of product so X Z into 1 would be X Z plus X into Y into Z complement. Now again this would be equal to X into Z plus applying the commutative law here we have X into Z complement into Y. Now we have another distributive law which is A plus B into CD whole is equal to A plus CD whole into A plus CD whole. Now applying this distributive law here we have A plus V into C and so this would be equal to A that is X Z plus B which is X Z complement the whole into A which is X Z plus C which is Y. Now for this expression we would use this distributive law and so this would be equal to X into Z plus Z complement the whole and this whole into X Z plus Y the whole. Now further we have X into 1 that is Z plus Z complement is 1 this into X Z plus Y the whole. Now X into 1 would be X into X Z plus Y the whole that is in place of X into 1 we have written X as A into 1 is equal to A. So again by the distributive law we have X into X Z plus X into Y so further this would be equal to X into Z plus X into Y using the idempotent law that is A into A is equal to A so X into X would be X only further using the distributive law this is equal to X into Z plus Y the whole. So the function F that we have found out equal to X into Y complement into Z plus X into Y into Z complement plus X into Y into Z is simplified as X into Z plus Y the whole. So this is the required circuit in which the buzzers Y and Z are parallel and X is in series with this parallel network. This completes the session hope you have understood the solution of this question.