Page 87 - Computer Science V2.0 Class 11

= (b ● b') ● a' (Using associative property)
= 0 ● a' (Using complementarity property)
= 0 (Using theorem 2 (ii))

Verify De Morgan's laws: (a ● b)' = a' + b'

3.4 Boolean Algebra for Logic Circuits

Digital logic circuits use Boolean algebra of the two-element set S = {0, 1}. The operations + and ● in the definition of
Boolean algebra are denoted by OR and AND respectively in logic circuits. The complement a' of an element a is also
denoted by NOT a in the logic circuits. While dealing with logic circuits, we shall use the operator symbols +, ●, ꞌ as
well as their equivalents OR, AND, NOT, respectively.

3.4.1 Boolean Functions

Boolean variables may be combined, using the basic Boolean operations NOT, AND, OR to give rise to complex Boolean
functions, also called Boolean expressions, such as x + y + z', xy' + zy' + (x + y)' z + tx', A +
BC' + D'B', x' + yz'. In a Boolean expression, NOT has the highest precedence, followed by AND and OR in the
decreasing order. Parentheses may be used to change the order of evaluation. Boolean variables x, y, z, t, ….,
and their complements x', y', z', t', …, respectively, are called literals. A Boolean function may be computed
for different values of variables using a truth table. For example, Table 3.7 shows computation of the Boolean function
F = x' + yz'.

x y z x' z' yz' F = x' + yz'
0 0 0 1 1 0 1
0 0 1 1 0 0 1
0 1 0 1 1 1 1

0 1 1 1 0 0 1
1 0 0 0 1 0 0
1 0 1 0 0 0 0
1 1 0 0 1 1 1
1 1 1 0 0 0 0

Table 3.7: Truth table for F = x' + yz'
Fig 3.4 shows an implementation of the Boolean function F = x' + yz' using two NOT gates, one OR gate, and one
AND gate.

x

F = x' + yz'
y

z

Fig 3.4: Logic Diagram for Boolean function F = x' + yz'

Boolean Logic 73