Absorption

The Absorption Laws

The absorption laws show that redundant terms can be "absorbed":

• A $\lor$ (A $\land$ B) = A (OR absorbs AND)
• A $\land$ (A $\lor$ B) = A (AND absorbs OR)

Intuitively: if you already have A, then adding "A and something" (or "A or something") doesn't change the result - A already determines it.

Simplifying with Absorption

Absorption patterns to recognize:

• A $\lor$ (A $\land$ B) = A
• (A $\land$ B) $\lor$ A = A (by commutativity)
• A $\land$ (A $\lor$ B) = A
• (A $\lor$ B) $\land$ A = A (by commutativity)

Look for a term that appears both alone AND inside a compound expression.

Absorption with Negation

There's a useful variant of absorption involving negation:

• A $\lor$ ($\neg$A $\land$ B) = A $\lor$ B
• A $\land$ ($\neg$A $\lor$ B) = A $\land$ B

The negated term "opens a door" for B when A is false (or true, respectively).

This is sometimes called the consensus or simplification law.

OR Absorption Practice

The OR absorption law: A $\lor$ (A $\land$ B) = A

If A is true, the whole expression is true (regardless of B).

If A is false, then (A $\land$ B) is also false, so the whole thing is false.

Either way, the result is just A.

AND Absorption Practice

The AND absorption law: A $\land$ (A $\lor$ B) = A

If A is false, the whole expression is false (regardless of B).

If A is true, then (A $\lor$ B) is also true, so we get true $\land$ true = true.

Either way, the result is just A.

Recognizing Absorption Patterns

Absorption applies when a term appears both:

1. By itself (outside)
2. Inside a compound expression (with AND or OR)

Patterns:

• A $\lor$ (A $\land$ B) = A ✓
• A $\land$ (A $\lor$ B) = A ✓
• A $\lor$ (B $\land$ C) = ? ✗ (no A inside - absorption doesn't apply)

Absorption vs Other Laws

Don't confuse absorption with:

Absorption requires A both outside AND inside a compound.

Why Absorption Works

Why does A $\lor$ (A $\land$ B) = A?

Case 1: A = True

• A $\lor$ (A $\land$ B) = T $\lor$ (T $\land$ B) = T $\lor$ B = T
• Result: T = A ✓

Case 2: A = False

• A $\lor$ (A $\land$ B) = F $\lor$ (F $\land$ B) = F $\lor$ F = F
• Result: F = A ✓

The B term becomes irrelevant - A determines everything!

Absorption in Complex Expressions

Absorption can simplify parts of larger expressions:

• (A $\lor$ (A $\land$ B)) $\land$ C = A $\land$ C (simplify inner part first)
• X $\lor$ (Y $\land$ (Y $\lor$ Z)) = X $\lor$ Y (absorption inside, then simplify)

Look for the pattern: term + (same term combined with something).

Recognizing Absorption Opportunities

Absorption patterns to recognize:

• P $\lor$ (P $\land$ Q) = P $\leftarrow$ P absorbs P $\land$ Q
• P $\land$ (P $\lor$ Q) = P $\leftarrow$ P absorbs P $\lor$ Q

Key insight: If one term is "contained in" another, the larger term can be absorbed.

Spotting it: Look for a variable that appears both alone AND in a compound term.

Multi-Step Absorption

Complex expressions may need multiple absorption steps:

1. First absorption reveals new patterns
2. Reorder to expose more opportunities
3. Apply absorption again
4. Combine with other laws as needed

Patience and systematic checking pays off!

Boolean Algebra Synthesis

Boolean Algebra Laws Summary:

Plus: Commutativity, Associativity, Distributivity, Double Negation

Boolean Algebra Challenge

Simplification strategy:

1. Look for complements (P $\land$ $\neg$P = F, P $\lor$ $\neg$P = T)
2. Apply identity/annihilation when constants appear
3. Look for idempotence (duplicate terms)
4. Look for absorption opportunities
5. Consider factoring or De Morgan to reveal patterns
6. Repeat until no more simplifications possible

Evaluating Absorption Expressions

The absorption laws state:

• P $\land$ (P $\lor$ Q) = P
• P $\lor$ (P $\land$ Q) = P

These may seem counterintuitive at first. Evaluating them with concrete values helps build understanding of why P always "absorbs" the rest.

Choosing the Right Law

Quick reference — signature patterns: