Simplifying Boolean Expressions

Simplifying Boolean expressions is a core skill in OCR H446. A simplified expression uses fewer operations and translates to a circuit with fewer gates, which means lower cost, less power consumption, and faster operation. This lesson brings together all the Boolean algebra laws to demonstrate systematic simplification.

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Why Simplify?

Before simplification	After simplification
More gates needed	Fewer gates needed
Higher power consumption	Lower power consumption
More complex circuit	Simpler circuit
Harder to debug	Easier to debug
Slower signal propagation	Faster signal propagation

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Strategy for Simplification

There is no single algorithm for simplifying Boolean expressions algebraically, but the following strategy works well:

1. Look for inverse pairs: A AND NOT A = 0, or A OR NOT A = 1
2. Look for idempotent terms: A AND A = A, or A OR A = A
3. Apply absorption: A OR (A AND B) = A
4. Factor common terms using the distributive law
5. Apply De Morgan's laws to simplify NOT over compound expressions
6. Apply identity and null laws to clean up constants (0 and 1)

Exam Tip: Always state the name of the law you are applying at each step. OCR mark schemes award marks for correct identification of laws.

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Worked Example 1

Simplify: (A AND B) OR (A AND NOT B) OR (NOT A AND B)

Step 1: Look at the first two terms: (A AND B) OR (A AND NOT B) Factor out A: A AND (B OR NOT B) Apply inverse law: B OR NOT B = 1 Apply identity law: A AND 1 = A

So far: A OR (NOT A AND B)

Step 2: Apply absorption law: A OR (NOT A AND B) = A OR B

Wait — let us verify this with the absorption variant. Actually, A OR (NOT A AND B) is a well-known identity: = (A OR NOT A) AND (A OR B) [distributive law: OR over AND] = 1 AND (A OR B) [inverse law] = A OR B [identity law]

Result: (A AND B) OR (A AND NOT B) OR (NOT A AND B) = A OR B

Verification:

A	B	Original expression	A OR B
0	0	(0 AND 0) OR (0 AND 1) OR (1 AND 0) = 0 OR 0 OR 0 = 0	0
0	1	(0 AND 1) OR (0 AND 0) OR (1 AND 1) = 0 OR 0 OR 1 = 1	1
1	0	(1 AND 0) OR (1 AND 1) OR (0 AND 0) = 0 OR 1 OR 0 = 1	1
1	1	(1 AND 1) OR (1 AND 0) OR (0 AND 1) = 1 OR 0 OR 0 = 1	1

Both columns match. Verified.

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Worked Example 2

Simplify: NOT(A OR B) OR (A AND B)

Step 1: Apply De Morgan's second law to NOT(A OR B): = (NOT A AND NOT B) OR (A AND B)

This cannot be simplified further using basic laws — it is already in a minimal Sum of Products form. Let us verify:

A	B	NOT A AND NOT B	A AND B	Result
0	0	1	0	1
0	1	0	0	0
1	0	0	0	0
1	1	0	1	1