Associative Laws of Set Theory

Theorem

Intersection is Associative

Set intersection is associative:

$A \cap \paren {B \cap C} = \paren {A \cap B} \cap C$

Union is Associative

Set union is associative:

$A \cup \paren {B \cup C} = \paren {A \cup B} \cup C$

Also defined as

Some sources include in the other identities, for example:

Symmetric Difference is Associative

Symmetric difference is associative:

$R \symdif \paren {S \symdif T} = \paren {R \symdif S} \symdif T$

Also known as

The are also known as the associative properties (of set theory).

Also see

• Commutative Laws of Set Theory
• Distributive Laws of Set Theory

Sources

• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): algebra of sets: $\text {(i)}$
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): algebra of sets: $\text {(i)}$