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= X.Y.Z' + X.Y'.Z' + X'.Y.Z'
= Σ(110, 100, 010)
OR
= Σ(6, 4, 2)
Alternatively, using the truth table method
X Y Z X+Y Z' X.Y+Y'.Z
0 0 0 0 1 0
0 0 1 0 0 0
0 1 0 1 1 1
0 1 1 1 0 0
1 0 0 1 1 1
1 0 1 1 0 0
1 1 0 1 1 1
1 1 1 1 0 0
Taking the minterms with 1 in their final column, we get
X'.Y.Z' + X.Y'.Z' + X.Y.Z'
OR
Σ(2, 4, 6)
1.11 DERIVATION OF A BOOLEAN EXPRESSION FROM
MULTIPLE LOGICAL STATEMENTS
A boolean expression is an algebraic form of the logical statements expressed in terms of Boolean variables and their
complements. Firstly, the truth table with all the possible input combinations is drawn. Then for each combination, the
output that the logical statements will produce is evaluated. Finally, the Boolean expression is derived either in SOP or
POS form according to the requirement of the problem.
Let us understand this better with some examples.
Example 1: The owner of a company pays a bonus to his salesmen as per the criteria given below.
a. If the salesman works overtime for more than 4 hours but does not work on off days/holidays.
OR
b. If the salesman works when festival sales are on and also updates showroom arrangements.
OR
c. If the salesman works on an off day/holiday when the festival sales are on.
The inputs are:
INPUTS
O Works overtime for more than 4 hours
F Festival sales are on
H Working on an off day/holiday
U Updates showroom arrangements
(In all the above cases 1 indicates yes and 0 indicates no)
Output: X [1 indicates yes, 0 indicates no in all cases]
35
Boolean Algebra 35

