Distributivity

AND distributes over OR and vice versa - the key to expanding and factoring Boolean expressions.

14 topics • ~1026 words

A valve A guards two parallel paths (B or C). Distributing A: put A in series with each path separately. Now you have (A-B) or (A-C)—same flow behavior, different schematic.

Distributivity lets you reshape expressions—expand a compact form into an explicit list of cases, or factor a verbose list back into something compact. It's the core move in converting between DNF and CNF, and in algebraic simplification.

AND Distributes over OR

Distributivity lets us "multiply out" one operation over another.

AND distributes over OR, just like multiplication distributes over addition:

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

Think of it like: a $\times$ (b + c) = (a $\times$ b) + (a $\times$ c)

OR Distributes over AND

A bypass valve A around a series section (B-C). Distributing: A becomes a bypass around B, AND a bypass around C. The (A or B) AND (A or C) structure captures: "either the bypass works, or that segment must be open."

Here's something special about Boolean algebra: OR also distributes over AND!

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

This is different from ordinary arithmetic, where addition does NOT distribute over multiplication: a + (b $\times$ c) $\neq$ (a + b) $\times$ (a + c) in general.

Boolean algebra has this "dual distributivity" property.

Applying Distributivity

Summary of distributivity laws:

  • A $\land$ (B $\lor$ C) = (A $\land$ B) $\lor$ (A $\land$ C) (AND over OR - expand)
  • (A $\land$ B) $\lor$ (A $\land$ C) = A $\land$ (B $\lor$ C) (AND over OR - factor)
  • A $\lor$ (B $\land$ C) = (A $\lor$ B) $\land$ (A $\lor$ C) (OR over AND - expand)
  • (A $\lor$ B) $\land$ (A $\lor$ C) = A $\lor$ (B $\land$ C) (OR over AND - factor)

Expanding AND over OR

To expand A $\land$ (B $\lor$ C), distribute A to each term inside:

A $\land$ (B $\lor$ C) = (A $\land$ B) $\lor$ (A $\land$ C)

Think of it like: a $\times$ (b + c) = ab + ac

Expanding OR over AND

To expand A $\lor$ (B $\land$ C), distribute A to each term inside:

A $\lor$ (B $\land$ C) = (A $\lor$ B) $\land$ (A $\lor$ C)

This is unique to Boolean algebra - in arithmetic, addition does NOT distribute over multiplication!

Factoring with AND

Factoring is distributivity in reverse. Look for common terms:

(A $\land$ B) $\lor$ (A $\land$ C) = A $\land$ (B $\lor$ C)

The common factor A is "pulled out" of both terms.

Factoring with OR

Factoring OR over AND:

(A $\lor$ B) $\land$ (A $\lor$ C) = A $\lor$ (B $\land$ C)

When the same term appears in multiple OR expressions connected by AND, we can factor it out.

Choosing Expand vs Factor

Expand when you want to "multiply out":

  • A $\land$ (B $\lor$ C) $\to$ (A $\land$ B) $\lor$ (A $\land$ C)

Factor when you see a common term:

  • (A $\land$ B) $\lor$ (A $\land$ C) $\to$ A $\land$ (B $\lor$ C)

Factoring often leads to simpler expressions.

Distributivity vs Arithmetic

In arithmetic:

  • a $\times$ (b + c) = (a $\times$ b) + (a $\times$ c) ✓ (multiplication over addition)
  • a + (b $\times$ c) $\neq$ (a + b) $\times$ (a + c) ✗ (addition does NOT distribute over multiplication)

In Boolean algebra:

  • A $\land$ (B $\lor$ C) = (A $\land$ B) $\lor$ (A $\land$ C) ✓ (AND over OR - like arithmetic)
  • A $\lor$ (B $\land$ C) = (A $\lor$ B) $\land$ (A $\lor$ C) ✓ (OR over AND - unique to Boolean!)

Factoring Practice

Factoring is the reverse of distribution:

  • (P $\land$ Q) $\lor$ (P $\land$ R) = P $\land$ (Q $\lor$ R) $\leftarrow$ factor out P
  • (P $\lor$ Q) $\land$ (P $\lor$ R) = P $\lor$ (Q $\land$ R) $\leftarrow$ factor out P

How to factor:

  1. Identify the common term
  2. Remove it from each part
  3. Combine what remains with the opposite operation

When to Expand vs Factor

When to expand:

  • Converting to DNF or CNF (standard forms)
  • Looking for complementary terms to cancel
  • Need to evaluate with specific values

When to factor:

  • Reducing the number of operations
  • Finding simpler equivalent forms
  • Exposing common structure

Neither is always better - choose based on your goal!

Multi-Term Distribution

Distributing across multiple terms:

P $\land$ (Q $\lor$ R $\lor$ S) = (P $\land$ Q) $\lor$ (P $\land$ R) $\lor$ (P $\land$ S)

Double distribution (FOIL-like):

(P $\lor$ Q) $\land$ (R $\lor$ S) = (P $\land$ R) $\lor$ (P $\land$ S) $\lor$ (Q $\land$ R) $\lor$ (Q $\land$ S)

This creates 2 $\times$ 2 = 4 terms. With more terms, it grows fast!

Strategic Distribution Challenge

The legacy spec reads: "Pump A active AND either backup online or backup offline." You reorganize the logic to expose the hidden redundancy: if either state of the backup satisfies the condition, the backup clause is always true—meaning only Pump A matters.

Strategic distribution means distributing in a way that reveals simplification opportunities:

  • Distribute to expose complements (P $\land$ $\neg$P)
  • Distribute to expose idempotence (P $\land$ P)
  • Distribute to expose absorption patterns

Don't blindly expand - look for what simplifications become possible!

Evaluating Distributive Expressions

The distributive laws in Boolean algebra are:

  • P $\land$ (Q $\lor$ R) = (P $\land$ Q) $\lor$ (P $\land$ R) (AND distributes over OR)
  • P $\lor$ (Q $\land$ R) = (P $\lor$ Q) $\land$ (P $\lor$ R) (OR distributes over AND)

Evaluating these expressions with concrete values shows why both forms always give the same result.

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