GATE CS 2020 Q.38 (CS Q.28): Boolean Algebra and Logic Minimization

Consider the Boolean function $z(a, b, c)$ represented by the logic circuit below.

Which one of the following minterm lists represents the circuit given above?

$z = \sum(1, 3, 7)$

$z = \sum(1, 4, 5, 6, 7)$

$z = \sum(2, 4, 5, 6, 7)$

$z = \sum(2, 3, 5)$

Official answer and solution

Solution

Derive the Boolean expression from the logic gate schematic and expand into minterms.

From the circuit diagram:

Input $b$ passes through an inverter, producing $\overline{b}$ .
An AND gate receives inputs $\overline{b}$ and $c$ , producing output $\overline{b} c$ .
An OR gate receives input $a$ and the output of the AND gate, producing: $z(a, b, c) = a + \overline{b} c$

Expand the terms into full minterm canonical sum:

Term $a$ $a$ : $a = a(b + \overline{b})(c + \overline{c}) = a b c + a b \overline{c} + a \overline{b} c + a \overline{b} \overline{c}$ $a = a (b + b) (c + c) = ab c + ab c + a b c + a b c$
- Minterms: $m_7 (111), m_6 (110), m_5 (101), m_4 (100)$ .
Term $\overline{b} c$ $b c$ : $\overline{b} c = (a + \overline{a})\overline{b} c = a \overline{b} c + \overline{a} \overline{b} c$ $b c = (a + a) b c = a b c + a b c$
- Minterms: $m_5 (101)$ (already included) and $m_1 (001)$ .

Union of minterms:

z(a, b, c) = \sum(1, 4, 5, 6, 7)

Final answer

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