Eary Logic ve means lements of a set of previousely ements of the Now the following principle tells us that if a statement is proved true then we have simultaneously proved that its dual is true If s is a theorem about a Theorem 2 The principle of duality Boolean algebra then so is its dual s It is because of this principle that the statements in Theorem 1 look so similar Let us now see how to apply Boolean algebra methods to circuit design For this purpose we shall introduce the necessary mathematical terminology and ideas in the following section 3 3 BOOLEAN EXPRESSIONS In Unit 2 you learnt how a compound statement can be formed by combining some propositions P1 P2 Pn say with the help of logical connectives A V and Analogously while expressing circuits mathematically we identify each circuit in terms of some Boolean variables Each of these variables represents either a simple switch or an input to some electronic switch Definition Let B X V A 0 1 be a Boolean algebra A Boolean expression in variables X1 X2 xk say each taking their values in the set X is defined recursively as follows i Each of the variables x x2 xk as well as the elements O and I of the Boolean algebra B are Boolean expressions ii If X and X are previously definecl Boolean expressions then X A X2 X V X2 and X are also Boolean expressions For instance x A x3 is a Boolean expression because so are x and x3 Similarly because x V x2 is a Boolean expression so is x1 V x2 A x AX3 If X is a Boolean expression in n variables x x2 Xn say we write this as X X X X Each variable x and its complement x 1 i k is called a literal For example in the Boolean expression X X1 X2 X3 x1 V x2 x1 X3 there are three literals namely X1 X2 and x3 In the context of designing a circuit or redesigning a circuit with fewer electronic switches we need to consider techniques for minimising Boolean expressions In the process we shall be using the concepts defined below Definition A Boolean expression in k variables x x2 xk is called a minterm if it is of the form y A y2 A Ayki i ii a maxterm if it is of the form y Vy2 V Vyki where each yj is a literal i e it is either an x or an x for 1 i k and Vi y for i j Thus a minterm or a maxterm in k variables is a meet or a join respectively of exactly k distinct variables For example X A x2 and x V x2 is a minterm a maxterm respectively in the two variables X and x2 Definition A Boolean expression involving k variables is in disjunctive