Absorption Law — Definition, Formula & Examples

The Absorption Law states that combining a variable with the conjunction (or disjunction) of itself and another variable simply returns the original variable. In short, the extra term gets 'absorbed' and has no effect on the result.

For any propositions $A$ and $B$ , the Absorption Law provides two identities: $A \lor (A \land B) = A$ and $A \land (A \lor B) = A$ . Equivalently, in set theory: $X \cup (X \cap Y) = X$ and $X \cap (X \cup Y) = X$ .

Key Formula

A \lor (A \land B) = A \qquad \text{and} \qquad A \land (A \lor B) = A

Where:

$A$ = Any proposition or Boolean variable
$B$ = Any proposition or Boolean variable

How It Works

When you see a logical expression where one variable appears both outside and inside a compound term, the Absorption Law lets you replace the entire expression with just that variable. For example, if you encounter

p \lor (p \land q)

during a proof or simplification, you can immediately write

p

. This works because the inner term (

p \land q

) can only be true when

p

is already true, so it adds nothing new. You can verify either absorption identity by constructing a truth table: in every row, the compound expression matches the value of the absorbed variable.

Worked Example

Problem: Simplify the logical expression

p \land (p \lor q)

using the Absorption Law.

Identify the pattern: The variable

p

appears both as the left operand and inside the disjunction

p \lor q

. This matches the second form of the Absorption Law:

A \land (A \lor B) = A

p \land (p \lor q)

Apply Absorption: Substitute

A = p

and

B = q

into the law to absorb the redundant term.

p \land (p \lor q) = p

Verify with a truth table: When

p

is false,

p \land (\text{anything})

is false. When

p

is true,

p \lor q

is true, so

p \land \text{true} = \text{true}

. In both cases the result equals

p

Answer: The expression simplifies to

p

Why It Matters

The Absorption Law is one of several identities you use repeatedly in discrete mathematics and digital logic design to simplify Boolean expressions. Reducing redundant terms leads to simpler circuit designs and shorter logical proofs, which matters in courses like AP Computer Science and college-level discrete math.

Common Mistakes

Mistake: Confusing the Absorption Law with the Idempotent Law (

A \lor A = A

) or the Domination Law (

A \lor \text{true} = \text{true}

Correction: Absorption specifically involves two distinct variables where one 'absorbs' a compound term containing both. Make sure you identify that the same variable appears both outside and inside the inner operation.