Write logic expressions from circuits or truth tables

What is Boolean Logic? 🤔

Think of Boolean logic like a set of rules for a light switch. The switch can only be in two positions: ON (1) or OFF (0). Boolean logic uses the same two values (0 and 1) to describe the behaviour of digital circuits.

The basic operations are:

• AND ($\land$) – both inputs must be 1 for the output to be 1.
• OR ($\lor$) – at least one input is 1 for the output to be 1.
• NOT ($\lnot$) – flips the value (0 becomes 1, 1 becomes 0).

From Circuits to Logic Expressions 🔌

Suppose you have a simple circuit with two inputs, A and B, and the following gates:

• A NOT gate (produces ¬A)
• A AND gate that takes ¬A and B as inputs
• An OR gate that combines the output of the AND gate with A

To write the logic expression, follow these steps:

1. Start with the innermost gate: ¬A ∧ B.
2. Take the output of that gate and OR it with the other input: (¬A ∧ B) ∨ A.

The final expression is: $F = (\lnot A \land B) \lor A$.

From Truth Tables to Logic Expressions 📊

A truth table lists every possible combination of inputs and the resulting output. From this table we can build a Sum of Products (SOP) expression.

1️⃣ Identify rows where the output F is 1. 2️⃣ For each of those rows, write a product (AND) term that matches the input values. 3️⃣ Combine all product terms with OR (∪) to get the SOP expression.

For the table above, the rows with output 1 are the 2nd, 3rd, 5th, and 8th rows. The corresponding product terms are:

• ¬A ∧ ¬B ∧ C
• ¬A ∧ B ∧ ¬C
• A ∧ ¬B ∧ ¬C
• A ∧ B ∧ C

Exam Tips for Writing Logic Expressions 📝

• Always label each gate or table row clearly.
• When converting a circuit, start from the input side and work towards the output.
• For truth tables, use the Sum of Products method first; if time allows, simplify using Karnaugh maps.
• Check your final expression by plugging in a few test values.
• Remember that ¬(A ∨ B) = ¬A ∧ ¬B (De Morgan’s law) – handy for simplifying.

Practice Problems 🧩

1. Given the circuit:
   • A OR gate with inputs B and C
   • The output of the OR gate goes into a NOT gate
   • The output of the NOT gate is then ANDed with A
   Write the logic expression for the final output.
2. From the truth table below, write the SOP expression:

   X	Y	Z	G
   0	0	0	0
   0	1	0	1
   1	0	1	1
   1	1	1	0

3. Convert the following Boolean expression to a simplified form using De Morgan’s law: $\lnot(A \lor \lnot B) \land C$