Cardinality of Sets: 9 Identities

Vocabulary flashcards covering the 9 cardinality identities of sets, De Morgan's laws, and memory rules for JEE set theory.

Complement Identity

$n(A') = n(U) - n(A)$, which states that $A'$ contains everything in $U$ that is not in $A$.

Union Identity

$n(A \cup B) = n(A) + n(B) - n(A \cap B)$, which adds $A$ and $B$ and subtracts their overlap once to prevent double-counting.

Disjoint Union Identity

$n(A \cup B) = n(A) + n(B)$, if $A \cap B = \emptyset$, because disjoint sets have no common elements to subtract.

Intersection with Complement Identity

$n(A \cap B') = n(A) - n(A \cap B)$, which counts elements in $A$ but not $B$ (the same as $A - B$).

Both Complements Identity

$n(A' \cap B') = n((A \cup B)') = n(U) - n(A \cup B)$, which counts everything outside $A \cup B$ using De Morgan's law.

Union of Complements Identity

$n(A' \cup B') = n((A \cap B)') = n(U) - n(A \cap B)$, which counts everything except the common part of $A$ and $B$ using De Morgan's law.

Difference Identity

$n(A - B) = n(A) - n(A \cap B)$, which removes the overlap $A \cap B$ from $A$.

Intersection Identity

$n(A \cap B) = n(A) + n(B) - n(A \cup B)$, which is obtained by rearranging the two-set union identity.

Three-Set Inclusion-Exclusion Identity

$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$, which adds the single sets, subtracts pairwise overlaps, and adds the triple overlap once.

Three-Set Inclusion-Exclusion Memory Pattern

• singles − pairs + triple