involving exactly k variables. For instance, the Boolean expression in 2 variables X(x1, x2) = (x^x2) V (xi ^x2) V (x Ax2) is in DNF because it is a join of three minterms, namely, x1 Ax2, x ^x2, and x ^x2, where each one of these involves exactly two variables. Observe that each minterm in a DNF should involve all the k variables in the expression X(x1, x2,...,xk), k> 2. For instance, the Boolean expression X(x1,x2, x3)=(x A x2) V (x A x2 A X3) is not in DNF because x A x2 is not a minterm of all the three variables. However, since we can write (x^x2) = (x A x) ^I = (x', A x2) A (x3 V x) = (x1 Ax2 Ax3) V (x) A x2 ^x3), (by Identity law) (by Complementation law) (by Distributive law) and so, the expression X(x1, x2.x3) with this change for x', A x2 is in the disjunctive normal form. Indeed, using similar techniques, any Boolean expression (0) can be written in disjunctive normal form. Let us work out an example to illustrate this technique. Example 4: Obtain a disjunctive normal form for the expression X(x1, x2, x3) = (x Ax2) V (x1 Ax3). Solution: We can write x Ax2 = ( 2) = (x A x2) A (x3 V x3) = (x Ax2 A x3) V (x; A X2 A x3) Also, X1 X3 = = (x A X3) A I (x1 A X3) A (x2 V X2) (x1 A x3 A x2) V (x A x3 Ax) = (x1 A X2 A x3) V (x1 A x2 A x3). (Identity law) (Complementation law) (Distributive law) (by Identity law) (by Complementation law) (by Distributive law) (by commutativity of A) Hence the required disjunctive normal form of the given expression X(x1,x2, x3) in three variables is given by (x A x2 A x3) V (x1 A x2 A x3) V (x1 A x2 A x3) V (x1 Ax2 A x3). Why don't you try an exercise now? ** * E2) Obtain the disjunctive normal form of the Boolean expression X(x1,x2, x3)=(x v x2)' V (x A x3). The conjunctive normal form is another important type of expression which is analogous to the concept of DNF. Definition A Boolean expression in k variables is said to be in conjunctive normal form (CNF, in short) if it is a meet of maxterms, each of which involves all the k variables. For instance, the Boolean expression X(x1, x2, x3) = (x V x2 Vx3) A (x V x V x3) A (x1 V x2 V x3), is in CNF because it is the meet of maxterms (x Vx2 Vx3), (x VxVx3) and (x V x2 V x3). Note that all 3 variables are involved in each maxterm. Boolean Al