Higher Engineering Mathematics

96 NUMBER AND ALGEBRA

Problem 2. Derive the Boolean expression and

construct a truth table for the switching circuit

shown in Fig. 11.6.

Figure 11.6

The parallel circuit 1 to 2 and 3 to 4 gives (A+B) and
this is equivalent to a single switching unit between
7 and 2. The parallel circuit 5 to 6 and 7 to 2 gives
C+(A+B) and this is equivalent to a single switch-
ing unit between 8 and 2. The series circuit 9 to 8
and 8 to 2 gives the output

Z=B·[C+(A+B)]

The truth table is shown in Table 11.3. Columns 1,
2 and 3 give all the possible combinations ofA,B
andC. Column 4 isBand is the opposite to column

2. Column 5 is theor-function applied to columns 1
   and 4, giving (A+B). Column 6 is theor-function
   applied to columns 3 and 5 givingC+(A+B). The
   output is given in column 7 and is obtained by apply-
   ing theand-function to columns 2 and 6, giving
   Z=B·[C+(A+B)].

Table 11.3

1 2 3 4 5 6 7

A B C B A+B C+(A+B) Z=B·[C+(A+B)]

0 0 0 1 1 1 0

0 0 1 1 1 1 0

0 1 0 0 0 0 0

0 1 1 0 0 1 1

1 0 0 1 1 1 0

1 0 1 1 1 1 0

1 1 0 0 1 1 1

1 1 1 0 1 1 1

Problem 3. Construct a switching circuit to

meet the requirements of the Boolean expres-

sion:Z=A·C+A·B+A·B·CConstruct the

truth table for this circuit.

The three terms joined byor-functions, (+), indicate

three parallel branches,

having: branch 1 AandCin series

branch 2 AandBin series

and branch 3 AandBandCin series

Figure 11.7

Hence the required switching circuit is as shown in

Fig. 11.7. The corresponding truth table is shown in

Table 11.4.

Table 11.4

1 2 3 4 5 6 7 8 9

A B C C A·C A A·B A·B·C Z=A·C+A·B

+A·B·C

0 0 0 1 0 1 0 0 0

0 0 1 0 0 1 0 0 0

0 1 0 1 0 1 1 1 1

0 1 1 0 0 1 1 0 1

1 0 0 1 1 0 0 0 1

1 0 1 0 0 0 0 0 0

1 1 0 1 1 0 0 0 1

1 1 1 0 0 0 0 0 0

Column 4 isC, i.e. the opposite to column 3

Column 5 isA·C, obtained by applying theand-

function to columns 1 and 4

Column 6 isA, the opposite to column 1

Column 7 isA·B, obtained by applying theand-

function to columns 2 and 6

Column 8 isA·B·C, obtained by applying the

and-function to columns 4 and 7

Column 9 is the output, obtained by applying the

or-function to columns 5, 7 and 8