Understanding these concepts of SOP and POS is crucial for anyone working with digital logic design.
Canonical and Standard Forms:
The logic function can be expressed in two standard forms:
1- Sum of Product (SOP):
The Sum of Products (SOP) is a form of expression in Boolean algebra where the expression is a sum (OR operation) of multiple product (AND operation) terms. Each product term comprises literals (variables or their complements) ANDed together.
2- Product of Sum (POS):
The Product of Sums (POS) is another form of expression in Boolean algebra where the expression is a product (AND operation) of multiple sum (OR operation) terms. Each sum term comprises literals (variables or their complements) ORed together.
To understand these concepts more clearly let's take examples:
Example 1: Find the canonical forms in SOP and POS for the following function:
F(x, y, z) = xy + x'y' + z'
= xy(z +z') + x'y'(z + z') + z'(x + x')(y + y')..........we multiply every term with that variable that does not exist
= xyz + xyz' + x'y'z +x'y'z' + z'(xy + xy' +x'y +x'y')
= xyz + xyz' + x'y'z + x'y'z' + xyz' + xy'z' + x'yz'........we remove the same term and just write one.
= xyz +xyz' + x'y'z + x'y'z' + xy'z' + x'yz'
F(SOP) = ∑ (0, 1, 2, 4, 6, 7)
These values represent the upper function in binary to decimal that's mean(nagation value = 0 and the value without negation = 1)
xyz = 111 ➔ 7
xyz' = 110 ➔ 6
x'y'z = 001 ➔ 1
x'y'z' = 000 ➔ 0
xy'z' = 100 ➔ 4
x'yz' = 010 ➔ 2
F(POS) = ∏ (3, 5) .....the values in POS are the values are not in the SOP
Example 2: Find the canonical forms in SOP and POS for the following function:
F = (A + B')(A + B + C')
= (A + B' + CC') (A + B + C')
= (A + B' + C) (A+ B '+ C') (A +B +C')
⭐In the POS we take a negation value equal to 1
Conclusion
The Sum of Products (SOP) and Product of Sums (POS) are two fundamental forms of representing Boolean functions in algebra. They play a crucial role in digital logic design, simplifying complex logical expressions and optimizing digital circuits. Mastering these concepts is required for anyone involved in the computer science field, electrical engineering, and mathematics.
Understanding and applying SOP and POS expressions will not only enhance your problem-solving skills but also provide a solid foundation for designing efficient and reliable digital systems.
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