Boolean Identities

Computer Science: OCR A Level, topic 4.3.7

Boolean Identities

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.

AND identities

A value AND false returns false.
$X. 0 = 0$
A value AND true returns the value.
$X.1 = X$
A value AND the same value returns the value.
$X.X = X$
A value AND NOT the same value returns false.
$X.\overline{X} = 0$

OR identities

A value OR false returns the value.
$X + 0 = X$
A value OR true returns true.
$X + 1 = 1$
A value OR the same value returns the value.
$X + X = X$
A value OR NOT the same value returns true.
$X +\overline{X} = 1$

Example

Using the identities, we can simplify the expression:
$(X.X)+(X+0)$
$(X . X) + X$
$X + X$
$X$

Test yourself on Boolean Identities

What is $Y.Y$ ?
What is $A+\overline{A}$ ?

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