Constructing Truth Tables

Truth tables are a systematic way to enumerate every possible combination of inputs and determine the resulting output for a Boolean expression or logic circuit. They are an essential tool for analysing, verifying, and comparing logic circuits and Boolean expressions in OCR H446.

---

Why Truth Tables Matter

Truth tables allow you to:

• Verify that a Boolean expression is correct
• Compare two expressions to see if they are logically equivalent
• Debug a logic circuit by checking expected outputs against actual outputs
• Prove Boolean algebra identities

---

How Many Rows?

The number of rows in a truth table depends on the number of input variables:

Variables	Rows (2^n)
1	2
2	4
3	8
4	16
5	32

Key Term: For n input variables, a truth table has 2^n rows. Each row represents a unique combination of 0s and 1s.

---

Standard Input Ordering

Inputs should be listed in a standard binary counting pattern. For three variables A, B, C:

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

This is simply counting from 0 (000) to 7 (111) in binary.

Exam Tip: Always use this standard ordering. Examiners expect it and it makes your work easier to check.

---

Order of Evaluation (Precedence)

When evaluating Boolean expressions, operations have a specific precedence (priority):

Priority	Operation	Symbol
1 (highest)	NOT	NOT, bar, '
2	AND	AND, ., *
3 (lowest)	OR	OR, +

Brackets override precedence, just as in normal arithmetic.

Example: Evaluate A OR B AND NOT C

1. First evaluate NOT C
2. Then evaluate B AND (NOT C)
3. Then evaluate A OR (B AND NOT C)

This is equivalent to A OR (B AND (NOT C)) — NOT binds tightest, then AND, then OR.

---

Step-by-Step Method for Building Truth Tables

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

Step 1: Identify the number of variables. Here: A, B (2 variables = 4 rows).

Step 2: List all input combinations.

Step 3: Break the expression into sub-expressions and create intermediate columns:

• Column 1: A OR B
• Column 2: A AND B
• Column 3: NOT(A AND B)
• Column 4: (A OR B) AND NOT(A AND B)

A	B	A OR B	A AND B	NOT(A AND B)	Q
0	0	0	0	1	0
0	1	1	0	1	1
1	0	1	0	1	1
1	1	1	1	0	0

Notice: Q = 1 when exactly one input is 1. This is the same as A XOR B. The truth table proves that (A OR B) AND NOT(A AND B) is equivalent to A XOR B.

---

Completing Truth Tables from Logic Circuits

When given a circuit diagram, you trace the signals through the gates:

1. Label every input wire (A, B, C, etc.)
2. For each gate, determine what expression it computes
3. Work from left to right (inputs to output)
4. Fill in intermediate values at each gate
5. The final gate produces the output Q

Worked Example:

Suppose a circuit has:

• Gate 1: A AND B (output = P)
• Gate 2: NOT C (output = R)
• Gate 3: P OR R (output = Q)

The overall expression is Q = (A AND B) OR (NOT C).