Distribution

Distribute conjunctions over disjunctions and vice versa

P∧(Q∨R) ≡ (P∧Q)∨(P∧R)
Understanding Distribution

Distribution laws show how logical operators can be distributed across each other, similar to how multiplication distributes over addition in arithmetic. These rules allow you to restructure logical expressions by moving conjunctions inside disjunctions and vice versa, which is essential for logical manipulation and proof construction.

Two Distribution Laws:

• P∧(Q∨R) ≡ (P∧Q)∨(P∧R) - Conjunction distributes over disjunction • P∨(Q∧R) ≡ (P∨Q)∧(P∨R) - Disjunction distributes over conjunction

Mathematical Analogy:

Think of it like arithmetic: a×(b+c) = (a×b)+(a×c). Similarly, logical AND distributes over OR, and OR distributes over AND, though the patterns are slightly different due to the nature of logical operators.

Symbolic Logic Examples
P∧(Q∨R) ≡ (P∧Q)∨(P∧R) and P∨(Q∧R) ≡ (P∨Q)∧(P∨R)

AND over OR Distribution

formal
Factored Form:
P∧(Q∨R)
Distributed Form:
(P∧Q)∨(P∧R)

OR over AND Distribution

formal
Factored Form:
P∨(Q∧R)
Distributed Form:
(P∨Q)∧(P∨R)

Complex Distribution

advanced
Complex Expression:
(A∨B)∧(C∨D∨E)
After Distribution:
((A∨B)∧C)∨((A∨B)∧D)∨((A∨B)∧E)

Key Point: Distribution allows you to expand or factor logical expressions, similar to algebraic distribution.

Examples & Applications

Example 1: Course Planning(Academic requirements)

beginner
Original statement:
I must take math and (physics or chemistry)
Distributed form:
(I must take math and physics) or (I must take math and chemistry)

Explanation: Distribution helps clarify academic requirements by showing all possible valid course combinations.

Example 2: Business Strategy(Market analysis)

intermediate
Strategic requirement:
We need marketing and (online sales or retail expansion)
Expanded options:
(We need marketing and online sales) or (we need marketing and retail expansion)

Explanation: In business contexts, distribution reveals distinct strategic paths while maintaining core requirements.

Example 3: System Architecture(Technical design)

advanced
System requirement:
Authentication required and (database or file storage)
Architecture options:
(Authentication and database) or (authentication and file storage)

Explanation: System design often requires distribution to enumerate all valid architectural configurations while preserving essential components.

Key Insights
Algebraic Similarity: Distribution in logic works similar to algebraic distribution: P∧(Q∨R) expands to (P∧Q)∨(P∧R), just like a(b+c) = ab+ac.
Two Directions: Distribution works both ways - you can expand (distribute) or factor (reverse distribute) logical expressions.
Bidirectional
Normal Forms: Essential for converting expressions to Conjunctive Normal Form (CNF) or Disjunctive Normal Form (DNF).
Proof Strategy: Often used in proofs to transform expressions into more workable forms for further manipulation.

Related Concepts

Understanding this concept connects to these important logical concepts: