Page 40 - Handout Digital Electronics

5.3 The Sum of Products (SOP) form

So, from the truth table the ORing (+) of input combinations where F =1, gives the sum of products
form (SOP).

The forms F = ABC ABC ABC ABC and F (A, B, C) = ∑ (0, 3, 5, 6).are called the sum of
products form. The two expressions above are most popular for expressing Boolean functions.

As already noted above, the sum of products form is derived from the truth table by noting where F = 1
and then writing the corresponding input combinations. If F = 1 in more than one instance the input
combinations are combined or joined by the OR (+) operator.

Activity

1 What is the meaning of the sum of products (SOP) in simple terms?

2 Show that sum of products (SOP) and product of sums (POS) are really the same

3 What is the real meaning of POS in operation terms?

5.4 The Product of Sums (POS) form
Dec A B C F
0 0 0 0 1
Consider the previous table 4 below. 1 0 0 1 0
2 0 1 0 0
From the truth table above, the product of sums 3 0 1 1 1
(POS) is formed by looking at output instances in 4 1 0 0 0
column F, where F is equal to 0 and then applying 5 1 0 1 1
group complementation and De Morgan’s 6 1 1 0 1
7 1 1 1 0 t
h
eorem. For example, F = 0 in the following input combinations, 001, 010,100,111. This can be
written as: F (A, B, C) ABC ABC ABC ABC . Since we rarely work with negative functions, the
above function must be converted to a positive function. This is done by using the complement law of
Boolean
algebra, that is A A . The above function can be written as:

F (A, B, C) F(A, B, C) ABC ABC ABC ABC . Applying De Morgan’s theorem, this can be

written as: F(A, B, C) (ABC ) (ABC ) (ABC ) (ABC) . Applying De Morgan theorem, this can
further be written as: F(A, B, C) (A B C) (A B C) (A B C) (A B C ) This last
expression is the product of sums. The product of sums (POS) simply means ANDing first and then
ORing as opposed to the sum of products (SOP) which means ORing first and then ANDing.

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