Writing Boolean Expressions from Truth Tables

A key skill for OCR J277 Section 2.5 is the ability to derive a Boolean expression from a truth table. This means looking at a truth table and writing the algebraic expression that describes when the output is 1. This lesson teaches you a systematic method for doing this.

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The Sum of Products Method

The most reliable method for deriving a Boolean expression from a truth table is called the sum of products (SOP) method. Here is the step-by-step process:

1. Identify all rows where the output is 1
2. For each of these rows, write a product term (AND expression) that describes the inputs
3. Combine all the product terms using OR

Rules for Writing Product Terms

• If an input is 1 in the row, write the variable as-is (e.g., A)
• If an input is 0 in the row, write the variable with NOT (e.g., NOT A)

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Worked Example 1: Two Inputs

Consider this truth table:

A	B	Q
0	0	0
0	1	1
1	0	1
1	1	0

Step 1: Identify rows where Q = 1:

• Row 2: A = 0, B = 1
• Row 3: A = 1, B = 0

Step 2: Write product terms:

• Row 2: (NOT A) AND B
• Row 3: A AND (NOT B)

Step 3: Combine with OR:

• Q = ((NOT A) AND B) OR (A AND (NOT B))

This is the Boolean expression for XOR — which makes sense, as the truth table matches XOR.

OCR Exam Tip: The sum of products method always works, even for complex truth tables. It may not give the simplest expression, but it will always give a correct one.

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Worked Example 2: Three Inputs

A	B	C	Q
0	0	0	0
0	0	1	1
0	1	0	0
0	1	1	1
1	0	0	0
1	0	1	0
1	1	0	1
1	1	1	1

Step 1: Rows where Q = 1: rows 2, 4, 7, 8.

Step 2: Product terms:

• Row 2 (A=0, B=0, C=1): (NOT A) AND (NOT B) AND C
• Row 4 (A=0, B=1, C=1): (NOT A) AND B AND C
• Row 7 (A=1, B=1, C=0): A AND B AND (NOT C)
• Row 8 (A=1, B=1, C=1): A AND B AND C

Step 3: Combine with OR:

• Q = ((NOT A) AND (NOT B) AND C) OR ((NOT A) AND B AND C) OR (A AND B AND (NOT C)) OR (A AND B AND C)