In 4—10 assume that B is a Boolean algebra with operations + and •. Prove each statement using only the axioms for a Boolean algebra and statements proved in the text or in lower-numbered exercises.
9. De Morgan’s law for +: For all a and b in B, .
Let be the Boolean algebra, with the operations, addition” ”and multiplication” ”
Let be the Boolean algebra, with the operations, addition and multiplication .
Suppose are any elements of .
Then, is the complement of is the complement or .
The uniqueness of the complement law tells that for all and in B, if and .
To show that it is enough to show that and, then use the uniqueness of complement law.
By the Distributive law for addition over multiplication .
By the Commutative law for addition
By the Associative law for addition
By the Complement law for addition
By the universal bound law for addition
By the identity law for multiplication
Therefore, for all
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