Exercises 12-15 provide an outline for a proof that the associative laws, which were included as axiom for a Bolllem algebre, can be derived from the other four axioms. The algebra, can be derived from the 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+.
Test for equality law: For all elements a,b,and c ns a Boolean algebra.
Without using the associative law, derive this law from the other four laws in the axioms for a Boolean algebra plus the result of exercise 12.
To derive the Test for equality using other four axioms of Boolean algebra instead of using associative law for .
The test for equality law states that for all elements , and in a Boolean algebra
If and then .
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 Test for equality law which is -If and then for all elements , and
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