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)

This expression is correct but can be simplified (covered in the next lesson).

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Going from Expression to Truth Table

You may also be asked to construct a truth table from a Boolean expression. The process is reversed:

1. List all possible input combinations (count in binary)
2. For each row, substitute the input values into the expression
3. Evaluate the expression to get the output

Example: Q = A AND (NOT B)

A	B	NOT B	A AND (NOT B)
0	0	1	0 AND 1 = 0
0	1	0	0 AND 0 = 0
1	0	1	1 AND 1 = 1
1	1	0	1 AND 0 = 0

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Using Boolean Algebra Notation

In the exam, you may see expressions written in formal notation:

English	Algebraic notation
A AND B	A . B or AB
A OR B	A + B
NOT A	Ā (bar over A)
A XOR B	A ⊕ B

So the expression (NOT A) AND B OR A AND (NOT B) could also be written as:

Q = ĀB + AB̄

OCR Exam Tip: In the exam, you can use either English (AND, OR, NOT) or algebraic notation (., +, bar). Be consistent within your answer. If you use algebraic notation, make sure your bars are clearly drawn over the correct variables.

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Truth Table to Expression: The Sum-of-Products Flow

The full SOP method is a fixed sequence: scan rows, write product terms, OR them. The flow below captures the routine you should follow on every "write an expression for this truth table" question.

flowchart TD
    TT[Truth table] --> SCAN[Scan every row]
    SCAN --> CHK{Is Q = 1?}
    CHK -->|No| NEXT[Skip - move to next row]
    CHK -->|Yes| BUILD[Build a product term: variable as-is for input 1, NOT variable for input 0]
    BUILD --> NEXT
    NEXT --> END{All rows checked?}
    END -->|No| SCAN
    END -->|Yes| OR[Combine all product terms with OR]
    OR --> SOP[Sum-of-products expression]
    SOP --> SIMP[Optional: simplify with Boolean laws]

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Practice Problem

Derive the Boolean expression from this truth table:

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

Solution:

• Row 1 (A=0, B=0): (NOT A) AND (NOT B)
• Row 3 (A=1, B=0): A AND (NOT B)
• Row 4 (A=1, B=1): A AND B

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

This simplifies to Q = (NOT B) OR A (which you can verify by checking all rows).

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Common Exam Mistakes

1. Missing a row — make sure you identify ALL rows where Q = 1, not just the first one
2. Incorrect NOT placement — NOT applies to inputs that are 0, not 1
3. Forgetting brackets — use brackets to make the order of operations clear
4. Not simplifying when asked — if the question says "simplify", the sum of products answer alone will not get full marks

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Summary

• The sum of products method derives a Boolean expression from a truth table
• Identify rows where Q = 1, write AND terms, combine with OR
• For inputs that are 0 in a row, use NOT; for inputs that are 1, use the variable directly
• You can also go from expression to truth table by substituting values
• Boolean algebra uses . for AND, + for OR, and a bar for NOT

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Extended Worked Example: 3-Input Scenario

A bike lock opens (Q = 1) when you enter the correct digit on at least two of three dials. Let A, B, C represent whether each dial is correct (1 = correct).

Build the truth table first by asking "is at least two of A, B, C equal to 1?":

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

Now apply sum of products. Q = 1 in four rows. Write a product term for each: