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Boolean algebra is the mathematical system used to analyse and simplify logic expressions. It provides a set of laws (rules) that allow you to manipulate and reduce expressions, just as algebraic laws allow you to simplify mathematical equations. Mastering these laws is essential for the OCR H446 exam.
In this lesson, we use the following notation:
| Symbol | Meaning |
|---|---|
| . or AND | Logical AND |
| + or OR | Logical OR |
| NOT A or A' | Logical NOT (complement) |
| 1 | TRUE (logic high) |
| 0 | FALSE (logic low) |
The identity laws state that combining a variable with the identity element leaves the variable unchanged.
| Law | Expression | Explanation |
|---|---|---|
| AND identity | A AND 1 = A | ANDing with 1 has no effect |
| OR identity | A OR 0 = A | ORing with 0 has no effect |
Example: If X = (A AND 1) OR 0, then X = A. Both operations leave A unchanged.
The null laws describe what happens when you combine a variable with the annihilating element.
| Law | Expression | Explanation |
|---|---|---|
| AND null | A AND 0 = 0 | ANDing with 0 always gives 0 |
| OR null | A OR 1 = 1 | ORing with 1 always gives 1 |
Example: No matter what A is, A AND 0 = 0. If one input to an AND gate is always 0, the output is always 0.
Combining a variable with itself gives the same variable.
| Law | Expression | Explanation |
|---|---|---|
| AND idempotent | A AND A = A | ANDing a value with itself gives the same value |
| OR idempotent | A OR A = A | ORing a value with itself gives the same value |
Example: A OR A OR A = A. Repeating an input in an OR expression is redundant.
Combining a variable with its complement gives a constant.
| Law | Expression | Explanation |
|---|---|---|
| AND inverse | A AND (NOT A) = 0 | A variable ANDed with its complement is always 0 |
| OR inverse | A OR (NOT A) = 1 | A variable ORed with its complement is always 1 |
Example: If A = 1, then NOT A = 0, so A AND NOT A = 1 AND 0 = 0. If A = 0, then NOT A = 1, so A AND NOT A = 0 AND 1 = 0. Either way, the result is 0.
NOT(NOT A) = A
Inverting a value twice returns the original value.
The order of operands does not matter.
| Law | Expression |
|---|---|
| AND commutative | A AND B = B AND A |
| OR commutative | A OR B = B OR A |
The grouping of operands does not matter.
| Law | Expression |
|---|---|
| AND associative | (A AND B) AND C = A AND (B AND C) |
| OR associative | (A OR B) OR C = A OR (B OR C) |
This means you can evaluate a chain of ANDs or a chain of ORs in any order.
These laws allow you to expand or factor expressions, similar to expanding brackets in algebra.
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