Absorption Laws

The absorption laws are logical rules that allow simplifying a compound proposition where a variable appears both outside and inside parentheses, and the main operators are different (a conjunction and a disjunction). They are called absorption laws because, in the resulting expression, one of the propositions appears to be "absorbed" or eliminated by the other, leaving a simpler formula.

There are two formulas:

• Law for conjunction: p ∧ (p ∨ q) ≡ p. 
• Law for disjunction: p ∨ (p ∧ q) ≡ p.

These are also known as total absorption laws, since one of the components disappears completely. The conditions to apply them are: a variable repeats outside and inside the parentheses, and the main connectives on each side of the parentheses are different.

These laws have an intuitive interpretation in everyday statements. For example, for the conjunction p ∧ (p ∨ q): "I am going to the grocery store and (I am going to the grocery store or to the pharmacy)." If the first part ("I am going to the grocery store") is true, the entire condition is satisfied regardless of the pharmacy, because the main action has already occurred. If the first part is false, the entire conjunction is false, making the second part irrelevant. The statement is absorbed and is equivalent to "I am going to the grocery store."

For the disjunction p ∨ (p ∧ q): "I study or (I study and work on practice problems)." If it is true that I study, the entire disjunction is true, absorbing the more specific condition. It would only be necessary to evaluate the conjunctive part if I do not study, but in that case the entire expression would be false. Therefore, the proposition simplifies to "I study."

Proofs

To prove that the equivalences of the absorption laws are valid, we use truth tables. 

1) Proof of p ∧ (p ∨ q) ≡ p:

p	q	p ∨ q	p ∧ (p ∨ q)
T	T	T	T
T	F	T	T
F	T	T	F
F	F	F	F

As shown, the final column p ∧ (p ∨ q) is identical to the initial column p, proving the logical equivalence. 

2) Proof of p ∨ (p ∧ q) ≡ p:

p	q	p ∧ q	p ∨ (p ∧ q)
T	T	T	T
T	F	F	T
F	T	F	F
F	F	F	F

Once again, the last column p ∨ (p ∧ q) matches that of p perfectly, confirming that the equivalence holds.

Partial Absorption Laws

The absorption laws also have alternative versions, known as partial absorption laws. These operate in a similar manner, but apply when the repeating variable appears negated inside the parentheses.

Their mechanism works as follows: given a proposition with a pattern where a variable repeats in both its simple and negated forms, the negated version inside the parentheses is "absorbed," and the entire expression reduces to a single connective—the main operator that was originally outside the parentheses.

The formulas are:

• For conjunction: p ∧ (¬p ∨ q) ≡ p ∧ q
• For disjunction: p ∨ (¬p ∧ q) ≡ p ∨ q

In the first case, the pattern is a conjunction (∧) outside parentheses containing a disjunction (∨). The negation ¬p disappears, leaving only the conjunction of p with q. An analogous process occurs in the second case.

To prove that the equivalences of the partial absorption laws are valid, we once again turn to truth tables.

We begin with the law for conjunction: p ∧ (¬p ∨ q) ≡ p ∧ q

p	q	¬p	¬p ∨ q	p ∧ (¬p ∨ q)	p ∧ q
T	T	F	T	T	T
T	F	F	F	F	F
F	T	T	T	F	F
F	F	T	T	F	F

The table shows that, regardless of the truth values of p and q, the final columns for both expressions match, confirming their logical equivalence.

The same applies to the second law: p ∨ (¬p ∧ q) ≡ p ∨ q

p	q	¬p	¬p ∧ q	p ∨ (¬p ∧ q)	p ∨ q
T	T	F	F	T	T
T	F	F	F	T	T
F	T	T	T	T	T
F	F	T	F	F	F

Applications

The absorption laws have a direct practical application in simplifying logical expressions. 

Example

Let us simplify the compound proposition (¬p ∨ q) ∧ [p ∨ (q ∧ p)].

First, apply the commutative property of conjunction inside the brackets:

(¬p ∨ q) ∧ [p ∨ (p ∧ q)] 

Apply the total absorption law for disjunction inside the brackets:

(¬p ∨ q) ∧ p 

Apply the commutative property of conjunction to the entire expression:

p ∧ (¬p ∨ q) 

Now apply the partial absorption law for conjunction:

p ∧ q

Therefore, the final simplification is p ∧ q.

Absorption in Set Theory

The absorption laws have a direct parallel in set theory, thanks to the natural correspondence between logical connectives and set operations. 

The two fundamental laws, analogous to total absorption, are formally expressed as follows:

A ∩ (A ∪ B) = A

A ∪ (A ∩ B) = A

For the first law, A ∩ (A ∪ B), the set A ∪ B contains all elements belonging to A and B. When intersecting it with A, any element that belongs only to B is excluded, so the result is simply A. For the second law, A ∪ (A ∩ B), the set A ∩ B is a subset of A—specifically the elements that A shares with B. Taking the union of this subset with all of A adds nothing new, yielding A once again.

There are also partial absorption versions, which involve the complement of a set, denoted as A'. The formulas are:

A ∩ (A' ∪ B) = A ∩ B

A ∪ (A' ∩ B) = A ∪ B

In the first case, A ∩ (A' ∪ B) = A ∩ B, the intersection of A with the union of its complement and B considers elements that must belong to A and, simultaneously, must belong either to A' or to B. However, no element can belong to A and its complement A' at the same time. Therefore, the condition of belonging to A' is impossible for elements of A to satisfy. This means that the only elements of A that can satisfy the second condition are those that also belong to B. The result is simply the intersection of A with B.

In the second case, A ∪ (A' ∩ B) = A ∪ B, the union of A with the intersection of its complement and B adds to A those elements of B that were not already in A, since A' ∩ B represents the elements that are in B but not in A. The final outcome is the complete union of A with B.