Canonical forms of a Boolean function
Boolean function of 2 to 4 variables and its normalised written forms: minterms and maxterms, first and second canonical forms, Σm and ΠM notations, from the table to the form and back.
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Boolean function: what is it?
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A map that sends each combination of n variables valued 0 or 1 to a value 0 or 1; its truth table has 2ⁿ rows.
A truth table defines it completely. Several expressions can describe the same function, which is why we normalise.
What is the canonical sum-of-minterms form of the function f below?
```text
a b | f
0 0 | 0
0 1 | 1
1 0 | 1
1 1 | 0`ā·b + a·b̄a·b + ā·b̄a + ba·b̄ + ā·b̄
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ā·b + a·b̄Take one minterm per row at 1: ā·b (0 1) and a·b̄ (1 0). a·b + ā·b̄ matches rows 00 and 11, which f does not set to 1.
A minterm equals 1 for … row of the table; a maxterm equals 0 for … row.
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A minterm equals 1 for a single row of the table; a maxterm equals 0 for a single row.
A product equals 1 only if every factor is 1; a sum equals 0 only if every term is 0. That fixes each variable, hence the row.
Minterm: what is it?
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A product containing every variable, each direct or complemented, which equals 1 for a single row of the table.
For the row a = 1, b = 0, c = 1, it is a·b̄·c: a variable at 0 is complemented, a variable at 1 stays direct.