Final answer:
To find the complement F' of the Boolean function F(x,y,z)=x'y + xyz', De Morgan's laws and Boolean algebra rules are applied to simplify to sum-of-products form. It can be shown that FF' always equals 0 and F + F' always equals 1 because F' is the complement of F.
Step-by-step explanation:
The Boolean function provided is F(x,y,z)=x'y + xyz'. To find the complement of this function, denoted as F', we need to apply De Morgan's laws and the complement rules for Boolean algebra.
- First, apply De Morgan's laws to get the complement of each term:
- F' = (x'y)'+(xyz')' = (x+y')(x'+y+z)
- Next, apply distributive laws to simplify:
- F' = (x'y)'+(xyz')' = (x'+y)(x'+y+z)
- Finally, apply Boolean algebra rules to further simplify:
- F' = x'y' + x'z + yz
Now, to show that FF' = 0 and F + F' = 1,
- FF' can be found by multiplying F and F'. As they are complements, they will not share any common terms when multiplied, resulting in a zero function.
- FF' = (x'y + xyz')(x'y' + x'z + yz) = 0
- F + F' is always 1 because it represents the sum of a function and its complement.
- F + F' = (x'y + xyz') + (x'y' + x'z + yz) = 1