Commutativity and Associativity

Order and grouping don't matter for AND and OR - rearrange operands freely.

12 topics • ~1016 words

Valve A in series with valve B, or valve B in series with valve A—the flow is identical either way. The order you arrange components in a line does not change the system's behavior.

Commutativity lets you rearrange terms freely—A $\land$ B is the same as B $\land$ A. Without this guarantee, you'd have to treat every ordering as a different expression, making simplification vastly more difficult. It also lets you line up matching terms for factoring and cancellation.

Commutativity

Commutativity means the order of operands can be swapped without changing the result:

A $\lor$ B = B $\lor$ A (OR is commutative)
A $\land$ B = B $\land$ A (AND is commutative)

Just like addition (3+5 = 5+3), you can rearrange the operands freely.

Associativity

Three valves in series: (A-B)-C or A-(B-C)—same line, same flow. The parentheses are just bookkeeping; the water does not care how you group the valves on paper.

Associativity means the grouping (parentheses) of operands can be changed without affecting the result:

(A $\lor$ B) $\lor$ C = A $\lor$ (B $\lor$ C) (OR is associative)
(A $\land$ B) $\land$ C = A $\land$ (B $\land$ C) (AND is associative)

This lets us write A $\lor$ B $\lor$ C or A $\land$ B $\land$ C without parentheses - the grouping doesn't matter!

Rearranging Expressions

Rearrange the valves on the schematic, regroup them with brackets—the flow diagram stays equivalent. What matters is which valves are in the system, not how you label them on the page.

Together, commutativity and associativity let us freely rearrange terms in an expression of all ANDs or all ORs:

A $\lor$ B $\lor$ C = B $\lor$ C $\lor$ A = C $\lor$ A $\lor$ B = ... (any order)
A $\land$ B $\land$ C = B $\land$ C $\land$ A = C $\land$ A $\land$ B = ... (any order)

This is useful for grouping related terms or matching patterns.

Commutativity Practice

Commutativity means operands can be swapped:

A $\lor$ B = B $\lor$ A
A $\land$ B = B $\land$ A

The operation (AND or OR) must stay the same.

Associativity Practice

Associativity means grouping (parentheses) can change:

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

All operations must be the same type (all AND or all OR).

Non-Commutative Operations

Not all operations are commutative. In Boolean logic:

Commutative:

AND: A $\land$ B = B $\land$ A ✓
OR: A $\lor$ B = B $\lor$ A ✓
XOR: A $\oplus$ B = B $\oplus$ A ✓
Equivalence: A $\leftrightarrow$ B = B $\leftrightarrow$ A ✓

NOT commutative:

Implication: A $\to$ B $\neq$ B $\to$ A (in general)

"If it rains, the ground is wet" $\neq$ "If the ground is wet, it rains"

Mixed Operations Warning

Important: Associativity only works when all operations are the same!

(A $\lor$ B) $\lor$ C = A $\lor$ (B $\lor$ C) ✓ (all OR)
(A $\land$ B) $\land$ C = A $\land$ (B $\land$ C) ✓ (all AND)
(A $\lor$ B) $\land$ C $\neq$ A $\lor$ (B $\land$ C) ✗ (mixed - NOT equal!)

When operations are mixed, changing parentheses changes the meaning!

Identifying Commutativity vs Associativity

Commutativity swaps operands: A $\lor$ B $\to$ B $\lor$ A

Associativity changes grouping: (A $\lor$ B) $\lor$ C $\to$ A $\lor$ (B $\lor$ C)

Key difference:

Commutativity: two terms, positions swap
Associativity: three+ terms, parentheses move

Flattening Expressions

Because of associativity, we can write chains of the same operation without parentheses:

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

This is called "flattening" - we can evaluate in any order.

When Reordering Helps

When to reorder terms:

To expose complements: P $\land$ Q $\land$ $\neg$P $\to$ P $\land$ $\neg$P $\land$ Q = F $\land$ Q = F
To expose idempotence: P $\lor$ Q $\lor$ P $\to$ P $\lor$ P $\lor$ Q = P $\lor$ Q
To enable absorption: P $\lor$ Q $\lor$ (P $\land$ R) $\to$ P $\lor$ (P $\land$ R) $\lor$ Q = P $\lor$ Q
To align for factoring: (P $\land$ Q) $\lor$ (P $\land$ R) - P is already aligned!

Reordering doesn't simplify by itself, but it enables other laws.

Non-Commutative Operations

Not all operations are commutative!

Operation	Commutative?	Example
AND, OR, XOR	Yes	P $\land$ Q = Q $\land$ P
Implication ($\to$)	No	P $\to$ Q $\neq$ Q $\to$ P
Subtraction	No	5 - 3 $\neq$ 3 - 5
Division	No	6 ÷ 2 $\neq$ 2 ÷ 6

Implication is the main non-commutative operation in logic!

Finding Equivalent Reorderings

Three valves in series: A, B, C. Check them in any order—ABC, BCA, CAB—the system either flows or it does not. The sequence of inspection does not change the underlying configuration.

With commutativity and associativity, many reorderings are equivalent:

For P $\land$ Q $\land$ R, all of these are equivalent:

P $\land$ Q $\land$ R
Q $\land$ P $\land$ R
R $\land$ Q $\land$ P
(P $\land$ Q) $\land$ R
P $\land$ (R $\land$ Q)
etc.

3 terms $\to$ 6 orderings. 4 terms $\to$ 24 orderings!

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