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 expression | As a switch | Comment |
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$A + 1 = 1$ (A OR 1 = 1) |
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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) |
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Only A is relevant as the other switch is open (i.e. 0) |
$A.1 = A$ (A AND 1 = A) |
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Only A controls the flow as the other switch is always closed (i.e. 1) |
$A.0 = 0$ (A AND 0 = A) |
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The position of A is irrelevant as the other switch is always open (i.e. 0) |
$A + A = A$ (A OR A = A) |
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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) |
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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) |
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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) |
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When one switch is open the other is closed, so current can never pass through |
Challenge see if you can find out one extra fact on this topic that we haven't already told you
Click on this link: boolean identity rules