What are the main set theory formulas?

The main set theory formulas include the commutative, associative, and distributive laws for union and intersection, the identity and domination laws involving the empty set and universal set, complement laws, De Morgan's laws, absorption laws, and cardinality formulas such as the inclusion-exclusion principle.

What are De Morgan's laws in set theory?

De Morgan's laws state that the complement of a union equals the intersection of the complements, and the complement of an intersection equals the union of the complements. Both laws generalize to arbitrary collections of sets.

How does the inclusion-exclusion principle work for two sets?

For two finite sets A and B, the cardinality of their union is |A| + |B| - |A intersection B|. Subtracting the intersection size corrects for elements that would otherwise be counted twice. When A and B are disjoint the formula simplifies to |A| + |B|.

What is the cardinality of a power set?

The power set of a set A with n elements contains exactly 2^n subsets. Each element is either included or excluded from a given subset, producing two independent choices per element and 2^n total subsets.

What is the difference between set difference and symmetric difference?

Set difference A minus B contains elements in A but not in B. Symmetric difference A triangle B contains elements in exactly one of the two sets, equivalent to the union minus the intersection. Set difference is not commutative while symmetric difference is.

Set Theory Formulas

28 formulas

Idempotent LawsGo to

Commutative LawsGo to

Associative LawsGo to

Distributive LawsGo to

Identity LawsGo to

Domination LawsGo to

Complement LawsGo to

De Morgan LawsGo to

Absorption LawsGo to

Cardinality FormulasGo to

Operation IdentitiesGo to

Set RelationshipsGo to

Idempotent Laws

(2 formulas)

Idempotent Law - Union

A \cup A = A

Idempotent Laws

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explanationrelated formulasrelated definitions

Combining a set with itself produces no new elements. The union absorbs duplicates by definition, so iterating the operation has no effect.

Idempotent Law - Intersection

A \cap A = A

Idempotent Laws

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explanationrelated formulasrelated definitions

The elements common to

A

and itself are exactly the elements of

A

. Repeated intersection collapses to a single set.

Commutative Laws

(2 formulas)

Commutative Law - Union

A \cup B = B \cup A

Commutative Laws

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explanationrelated formulasrelated definitions

The order of operands does not change the result of a union. Membership in

A \cup B

is symmetric in

A

and

B

— an element belongs if it is in either set.

Commutative Law - Intersection

A \cap B = B \cap A

Commutative Laws

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explanationrelated formulasrelated definitions

Order is irrelevant for intersection. The condition "

x \in A

and

x \in B

" is logically symmetric.

Associative Laws

(2 formulas)

Associative Law - Union

(A \cup B) \cup C = A \cup (B \cup C)

Associative Laws

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explanationrelated formulasrelated definitions

Grouping is irrelevant when unioning three or more sets. Both sides equal the set of elements belonging to at least one of

A

B

C

. Associativity justifies writing

A \cup B \cup C

without parentheses.

Associative Law - Intersection

(A \cap B) \cap C = A \cap (B \cap C)

Associative Laws

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explanationrelated formulasrelated definitions

Both sides equal the set of elements belonging to all three of

A

B

C

. Allows the unambiguous notation

A \cap B \cap C

Distributive Laws

(2 formulas)

Distributive Law - Intersection over Union

A \cap (B \cup C) = (A \cap B) \cup (A \cap C)

Distributive Laws

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explanationderivationrelated formulasrelated definitions

Intersection distributes over union. To intersect

A

with a union, intersect

A

with each piece separately and combine. Mirrors multiplication distributing over addition in arithmetic.

Distributive Law - Union over Intersection

A \cup (B \cap C) = (A \cup B) \cap (A \cup C)

Distributive Laws

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explanationderivationrelated formulasrelated definitions

Union distributes over intersection. Unlike arithmetic, set algebra has full duality — both operations distribute over each other. Swapping

\cup \leftrightarrow \cap

in any identity yields another identity.

Identity Laws

(2 formulas)

Identity Law - Union

A \cup \emptyset = A

Identity Laws

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explanationrelated formulasrelated definitions

The empty set is the identity element for union. Adding "nothing" to

A

leaves

A

unchanged.

Identity Law - Intersection

A \cap U = A

Identity Laws

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explanationconditionsrelated formulasrelated definitions

The universal set is the identity element for intersection. Restricting

A

to "everything" leaves

A

unchanged.

Domination Laws

(2 formulas)

Domination Law - Union

A \cup U = U

Identity and Domination Laws

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explanationconditionsrelated formulasrelated definitions

The universal set absorbs any union — adding any subset to

U

cannot exceed

U

. The dual of

A \cap \emptyset = \emptyset

Domination Law - Intersection

A \cap \emptyset = \emptyset

Identity and Domination Laws

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explanationrelated formulasrelated definitions

The empty set absorbs any intersection — there can be no element common to

A

and a set with no elements. The dual of

A \cup U = U

Complement Laws

(3 formulas)

Complement Law - Union

A \cup A^c = U

Complement Laws

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explanationconditionsrelated formulasrelated definitions

Every element of

U

either belongs to

A

or fails to belong to

A

— the law of excluded middle expressed in set form. Together

A

and its complement cover the universe.

Complement Law - Intersection

A \cap A^c = \emptyset

Complement Laws

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explanationrelated formulasrelated definitions

No element can simultaneously belong to

A

and not belong to

A

— the law of non-contradiction expressed in set form.

Double Complement Law

(A^c)^c = A

Complement Laws

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explanationderivationrelated formulasrelated definitions

Negating membership twice returns the original condition. The complement operation is its own inverse — an involution.

De Morgan Laws

(2 formulas)

De Morgan Law - Union

(A \cup B)^c = A^c \cap B^c

De Morgan Laws

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explanationderivationvariantsrelated formulasrelated definitions

The complement of a union is the intersection of complements. To not belong to

A \cup B

, an element must fail to belong to both

A

and

B

separately.

De Morgan Law - Intersection

(A \cap B)^c = A^c \cup B^c

De Morgan Laws

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explanationderivationvariantsrelated formulasrelated definitions

The complement of an intersection is the union of complements. To not belong to

A \cap B

, an element need only fail to belong to one of

A

B

Absorption Laws

(2 formulas)

Absorption Law - Union

A \cup (A \cap B) = A

Absorption Laws

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explanationderivationrelated formulasrelated definitions

Adding

A \cap B

A

contributes nothing new — every element of

A \cap B

already lies in

A

. The inner intersection is "absorbed" by the outer union.

Absorption Law - Intersection

A \cap (A \cup B) = A

Absorption Laws

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explanationderivationrelated formulasrelated definitions

Restricting

A \cup B

to elements also in

A

recovers exactly

A

. The inner union is "absorbed" by the outer intersection.

Cardinality Formulas

(6 formulas)

Inclusion-Exclusion - Two Sets

|A \cup B| = |A| + |B| - |A \cap B|

Cardinality - Finite Sets

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explanationconditionsvariantsrelated formulasrelated definitions

Counting

A

and

B

separately double-counts elements in

A \cap B

. Subtracting the intersection corrects for the overlap.

Inclusion-Exclusion - Three Sets

|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C|

Inclusion-Exclusion Principle

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explanationconditionsvariantsrelated formulasrelated definitions

Extends the two-set formula. Pairwise intersections are subtracted to undo double-counting; the triple intersection is added back because it was subtracted three times after being added three times.

Cardinality of Disjoint Union

|A \cup B| = |A| + |B| \quad \text{when } A \cap B = \emptyset

Cardinality of Disjoint Union

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explanationconditionsvariantsrelated formulasrelated definitions

When sets share no elements, the size of the union is simply the sum of the sizes. Special case of inclusion-exclusion with

|A \cap B| = 0

Cardinality of Power Set

|\mathcal{P}(A)| = 2^{|A|}

Power Set

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explanationderivationrelated formulasrelated definitions

For each element of

A

, a subset either includes it or excludes it — two independent choices per element. With

n

elements this yields

2^n

subsets.

Number of Proper Subsets

\text{number of proper subsets of } A = 2^{|A|} - 1

Number of Subsets

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explanationconditionsrelated formulasrelated definitions

Among the

2^n

subsets of an

n

-element set, exactly one is the set itself — excluded from the proper subsets. The remaining

2^n - 1

are proper.

Cantor Theorem

|A| < |\mathcal{P}(A)|

Comparing Cardinalities

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explanationderivationrelated formulasrelated definitions

The power set is always strictly larger than the original set, finite or infinite. Implies an unbounded hierarchy of infinite cardinalities — there is no largest set.