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أستاذ المادة انتهاء عبد الله محمد       16/06/2021 15:55:17
Boolean Expressions For Truth Table
All Boolean expressions, regardless of their form, can be converted into either
of two standard forms: the sum-of-products form or the product-of-sums form.
Standardization makes the evaluation, simplification, and implementation of
Boolean expressions much more systematic and easier.
4.6.1 The Sum-of-Products (SOP) Form (Minterm)
This form is sometimes called "minterm". A product term that contains each
of the n-variables factors in either complemented or uncomplemented form
for output digits "1" only, is called SOP. For example for the truth table below:
Input Output
A B C F
0 0 0 1 ?????????
0 0 1 0 ????????
0 1 0 1 ????????
0 1 1 1 ???????
1 0 0 0 ????????
1 0 1 0 ???????
1 1 0 1 ???????
1 1 1 1 ??????
The Logical SOP expression for the output digit "1" is written as"
?? = ????????? + ???????? + ??????? + ??????? + ??????
This function com be put in another form such as:
?? = ? 0, 2,3,6,7
Since F= 1 in rows 0, 2,3,6,7 only.
The second form is called the Canonical Sum of Products (Canonical SOP).68
4.6.2 The Product-of-Sum (POS) Form (Maxterm)
A Logical equation can also be expressed as a product of sum (POS) form
(sometimes this method is called "Maxterm". This is done by considering the
combination for F=0 (output = 0).
So for the above example from the truth table F=0 is in rows 1, 4, 5 hence:
???(??, ??, ??) = ???????? + ???????? + ???????
??(??, ??, ??) = ??? ?(??, ??, ??) = ???????????????+????????????????? ?+???????????????
= ?? ???????????? ? ?? ???????????? ? ?? ???????????
= (??? + ??? + ???) ? (??? + ??? + ???) ? (??? + ??? + ???)
??(??, ??, ??) = (?? + ?? + ???) ? (??? + ??? + ???) ? (??? + ??? + ???)
This is POS form. POS form can be expressed as:
?? = ? 1, 4, 5
This form is called the Canonical Product of Sum (Canonical POS).
Example: Put F in SOP and POS form and simplifying it:
Sol.
??????: ??(??, ??) = ? 0,1,3
= ?????? + ????? + ????
= ??? (??? + ?? ) + ????
= ??? + ????

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