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 universal bound law for + states that for every element a in a Boolean algebra, The proof shown in exercise 2 used the associative law for + . Rederive the law without using the associative law and using only the other four axioms for a Boolean algebra.
To derive the universal bound law for using other four axioms of Boolean algebra instead of using associative law for .
The universal bound law for states that for every element 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 universal bound law for which is , for every element
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