Exercises 12-15 provide an outline for a proof that the associative laws, which were included as an axiom for a Boolean algebra, can be derived from the other four axioms. The outline is from Introduction to Boolean Algebra by S. Givant and P. Halmos, Springer, 2009. In order to avoid unneeded parentheses, assume that. Takes precedence over +.
The absorption law for + states that for all elements a and b in a Boolean algebra, . Prove this law without using the associative law and using only the other four axioms for a Boolean algebra plus the result of exercise 12.
To derive the absorption law for using other four axioms of Boolean algebra instead of using associative law for .
The absorption law for states that for all elements and in a Boolean algebra .
There are total five axioms of the Boolean algebra which are listed in the following table.
|(1) Commutative laws|
|(2) Associative laws|
|(3) Distributive laws|
|(4) Identity laws|
|(5) Complement laws|
Now, we have to prove the absorption law for which is , for all elements and .
Consider the left hand side of the equation and apply the following axioms in order −
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