A Boolean function is described by an algebraic expression consisting of binary variables, the constants 0 and 1, and the logic operation symbols
Canonical and Standard Forms – Any binary variable can take one of two forms,
Relation between Minterms and Maxterms – Each minterm is the complement of it’s corresponding maxterm. For example, for a boolean function in two variables –
[Tex](m_0)^\prime=(x^\prime y^\prime)^\prime[/Tex] [Tex](m_0)^\prime=x+y=M_0[/Tex]In general or
Constructing Boolean Functions – Now that we know what minterms and maxterms are, we can use them to construct boolean expressions. “A Boolean function can be expressed algebraically from a given truth table by forming a minterm for each combination of the variables that produces a 1 in the function and then taking the OR of all those terms.” For example, consider two functions
x | y | z | f 1 | f 2 |
---|---|---|---|---|
0 | 0 | 0 | 0 | 0 |
0 | 0 | 1 | 1 | 0 |
0 | 1 | 0 | 0 | 0 |
0 | 1 | 1 | 0 | 1 |
1 | 0 | 0 | 1 | 0 |
1 | 0 | 1 | 0 | 1 |
1 | 1 | 0 | 0 | 1 |
1 | 1 | 1 | 1 | 1 |
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Example 1 – Express the following boolean expression in SOP and POS forms-
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Solution – The expression can be transformed into SOP form by adding missing variables in each term by multiplying by
where is the missing variable. It follows from the fact that – [Tex]F=x(y+y^\prime)+y^\prime z\\ F=xy+xy^\prime+y^\prime z\\ F=xy(z+z^\prime)+xy^\prime(z+z^\prime)+(x+x^\prime)y^\prime z\\ F=xyz + xyz^\prime + xy^\prime z + xy^\prime z^\prime + xy^\prime z + x^\prime y^\prime z [/Tex]On rearranging the minterms in ascending order If we want the POS form, we can double negate the SOP form as stated above to get- The SOP and POS forms have a short notation of representation-
Standard Forms – Canonical forms are basic forms obtained from the truth table of the function. These forms are usually not used to represent the function as they are cumbersome to write and it is preferable to represent the function in the least number of literals possible. There are two types of standard forms –
- Sum of Products(SOP)- A boolean expression involving AND terms with one or more literals each, OR’ed together.
- Product of Sums(POS) A boolean expression involving OR terms with one or more literals each, AND’ed together, e.g.
SOP-POS-
- Note – The above expressions are not equivalent, they are just examples.
GATE CS Corner Questions
Practicing the following questions will help you test your knowledge. All questions have been asked in GATE in previous years or in GATE Mock Tests. It is highly recommended that you practice them. 1. GATE CS 2010, Question 6 2. GATE CS 2008, Question 7 3. GATE CS 2014 Set-1, Question 17
References-
Digital Design 5th Edition, by Morris Mano and Michael Ciletti