E7) Let B={0, 1} consist of the bits 0 and 1. Show that B is a Boolean algebra, i.e., that the bits 0 and 1 form a two-element Boolean algebra. As said before, a logic circuit can be designed using elementary gates, where the output from an AND-gate, or an OR-gate, or a, NOT-gate is used as an input to other such gates in tlie circuitry. The different levels of voltage in these circuits, starting from the input lines, move only in the direction of the arrows as shown in all the figures given below. For instance, one combination of tlie three elementary gates is shown in Fig.9. Boolean A Fig. 9: A logic circuit of elementary gates. Now let us try to see the connection between logic circuits and Boolean expressions. We first consider the elementary gates. For a given pair of inputs x and x2, the output in the case of each of these gates is an expression of the form X A X2 or x V X2 or x'. Next, let us look at larger circuits. Is it possible to find an expression associated with a logic circuit, using the symbols A, V and '? Yes, it is. We will illustrate the technique of finding a Boolean expression for a given logic circuit with tlie help of some examples. But first, note that the output of a gate in a circuit may serve as an input to some other gate in the circuit, as in Fig. 9. So, to get an expression for a logic circuit the process always moves in the direction of the arrows in the circuitry. With this in mind, let us consider some circuits. Example 7: Find the Boolean expression for the logic circuit given in Fig.9. above. Solution: In Fig.9, there are four input terminals. Let us call them X1, X2, X3 and x. So, x and x2 are inputs to an OR-gate, wliicli gives X V x2 as an output expression (see Pig. 9(a)). Similarly, the other two inputs x3 arid x4, are inputs to an AND-gate, They will give x3 A x as an output expression. This is, in turn, an input for a NOT-gate in the circuit. So, this yields (x3 Ax4)' as the output expression. Now, both the expressions x V x2 and (x3 Ax4)' are inputs to the extreme right AND-gate in the circuit. So, they give (x1 V x2) A (x3 Ax4)' as the final output expression, which represents the logic circuit. X1 X2 D X1V X2 X3 X3 X4 (X3 ^x4) x4 Fig. 9(a) (x1 VX2) ^ (x^x)'