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 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 x 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 1 Solution In Fig 9 there are four input terminals Let us call themi X1 X2 X3 and x4 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 x4 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 x V x2 x3 x as the final output expression which represents the logic circuit X1 X X3 X1 V X X3 X4 X3 X4 Fig 9 a x V X X X Boolean A