2. Boolean Identities

The first step in simplifying a Boolean expression using algebra is to establish how inputs interact with one another.

You can learn this kind of thing by rote, just remembering that "A + 1 = 1", etc. But these rules can seem fairly abstract. So a better way of remembering them is to imagine Boolean operations as a group of switches in an electrical circuit.

For example, an AND operation is a pair of switches in series (one after another)

Current can only pass through if both switch A AND switch B are closed.

Similarly, an OR switch can be imagined as a pair of switches in parallel, where current can pass through either one:

These rules will be useful when simplifying expressions.

Boolean Identities

Boolean expression	As a switch	Comment
$A + 1 = 1$ (A OR 1 = 1)		If one switch is always closed (i.e. 1), the result doesn't change whether A is open or closed
$A + 0 = A$ (A OR 0 = A)		Only A is relevant as the other switch is open (i.e. 0)
$A.1 = A$ (A AND 1 = A)		Only A controls the flow as the other switch is always closed (i.e. 1)
$A.0 = 0$ (A AND 0 = A)		The position of A is irrelevant as the other switch is always open (i.e. 0)
$A + A = A$ (A OR A = A)		If both switches use the same input, they open and close together. It's the same as having just one switch. This is called 'idempotence'
$A.A = A$ (A AND A = A)		If both switches use the same input, they open and close together. It's the same as having just one switch. This is another example of 'idempotence'
$NOT \overline A = A$ (NOT NOT-A = A)		Double negatives cancel out
$A + \overline A = 1$ (A OR NOT-A = 1)		If switches use opposite inputs, one is always open when the other is closed, so current can always pass through.
$A.\overline A = 0$ (A AND NOT-A = 0)		When one switch is open the other is closed, so current can never pass through