Set Theory & Venn Diagrams

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medium importance~1 Q in Tier 124 formulas⚡ 15 shortcuts5 subtopics

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Every formula in this topic, grouped by subtopic. Print it and pin it above your desk.

Sets, Subsets & Power Sets

Number of subsets

2^n

A set with $n$ elements; the power set has $2^n$ elements.

Proper subsets

2^n - 1

All subsets except the set itself.

Non-empty subsets

2^n - 1

All subsets except the empty set.

Non-empty proper subsets

2^n - 2

Drops both the empty set and the full set.

Subsets of size r

^{n}C_r

Order does not matter inside a subset.

Union of two sets

n(A \cup B) = n(A) + n(B) - n(A \cap B)

Subtract the overlap counted twice.

De Morgan's laws

(A \cup B)' = A' \cap B',\quad (A \cap B)' = A' \cup B'

A complement swaps the join sign.

Two-Set Venn Diagrams

Union

n(A \cup B) = n(A) + n(B) - n(A \cap B)

At least one of the two groups.

Only A

n(A) - n(A \cap B)

The wing region outside B.

Exactly one

n(A) + n(B) - 2\,n(A \cap B)

Only A plus only B.

Neither

n(U) - n(A \cup B)

Outside both circles.

Three-Set Venn Diagrams

Union of three sets

n(A \cup B \cup C) = n(A) + n(B) + n(C) - p_{AB} - p_{BC} - p_{CA} + t

$p$ are the pair intersections and $t$ the triple.

Exactly two

p_{AB} + p_{BC} + p_{CA} - 3t

Petals only; the triple is removed from each pair.

At least two

p_{AB} + p_{BC} + p_{CA} - 2t

Exactly two plus the triple.

Exactly one

n(A) + n(B) + n(C) - 2(p_{AB} + p_{BC} + p_{CA}) + 3t

The three wings together.

Maximum–Minimum Region Problems

Maximum overlap

\max n(A \cap B) = \min(n(A), n(B))

One set wholly inside the other.

Minimum overlap

\min n(A \cap B) = \max(0,\ n(A) + n(B) - N)

$N$ is the size of the universal set.

Maximum neither

N - \max(n(A), n(B))

Shrink the union to its largest member set.

Minimum triple

\max(0,\ n(A) + n(B) + n(C) - 2N)

Union fixed at $N$; everyone covered.

Maximum triple

\min(n(A), n(B), n(C))

All three sets nested.

Survey & Caselet Word Problems

Union of two groups (percent)

(A \cup B)\% = a\% + b\% - (A \cap B)\%

Same formula, percent units.

Neither (percent)

100 - (a + b - ab)

$ab$ is the both percent here.

Union of three groups (percent)

(A \cup B \cup C)\% = a + b + c - p_{AB} - p_{BC} - p_{CA} + t

Inclusion–exclusion in percent units.

Exactly one (percent)

a + b - 2ab

Two groups; percent units.