Boolean Rules

Computer Science: OCR A Level, topic 4.3.8

Rules of Boolean Operations

There are a number of rules that can be used to simplify boolean expressions. Simplified expressions are cheaper and easier to implement as fewer logic gates are needed.

Absorption

Absorption

We can simplify expressions in the following way:
$X+(X.Y) = X +X = X$
We can only do this if:
The value outside the brackets must be present within the brackets.
The operators inside and outside the brackets must be different.
We say that Y has been absorbed.

Commutative

Commutative

The commutative rule states that values in AND or OR expressions can be swapped without changing the result:
$X. Y = Y . X$
$X + Y = Y + X$

Distributive

Distributive

A value outside of a set of brackets can be distributed between the values within the brackets:
$X . (Y + Z) = (X . Y) + (X . Z)$
$X + (Y . Z) = (X + Y) .(X + Z)$

Associative

Associative

The associative rule confirms that the order of operators in an expression has no impact on the result:
$(X + Y) + Z = X + (Y + Z)$
$(X . Y) .Z= X . (Y . Z)$

Double Negation

Double Negation

The equivalent of any double-negative is a positive:
$\overline{(\overline{X})} = X$

De Morgan's Law

De Morgan's Law

De Morgan's law is as follows:
$\overline{X + Y} = \overline{X} . \overline{Y}$
$\overline{X . Y} = \overline{X} + \overline{Y}$
We can use this law to convert part of an expression so the whole expression only uses one type of operator.
This reduces the cost as only one type of gate is required.

Test yourself on Boolean Rules

Which law allows us to simplify the expression shown:
$X+(X.Y) = X +X$
Use the distributive rule to simplify:
$A . (B + C)$
Which rule confirms that the order of operators in an expression has no impact on the result?

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