Venn Diagrams & Set Reasoning

A Venn diagram uses overlapping regions to show membership in groups. The overlap counts members in both groups.

Start with the deepest overlap, then fill exactly-two and only-one regions. Keep "at least" separate from "exactly".

Ten pupils like chess, eight like drawing and three like both. How many like at least one? 1. Adding 10 and 8 counts the three shared pupils twice. 2. Subtract one copy of the overlap. 3. 10 + 8 − 3 = 15 pupils like at least one.

The shared region contains the three pupils who like both activities. Subtract that overlap from each group total: chess only is 10 − 3 = 7 and drawing only is 8 − 3 = 5. Add the three disjoint regions to obtain 15; the diagram gives no count for pupils who like neither.

Among 100 people, 50 study A, 40 B, 30 C; pairwise overlaps AB=20, AC=15, BC=10 include the 5 studying all three. Union=50+40+30-20-15-10+5=80; none=20. Exactly two=(20-5)+(15-5)+(10-5)=30. Only A=50-20-15+5=20. Summing disjoint regions checks the result.

“At least one” includes the overlap. “Exactly one” excludes it. Pairwise overlaps usually include the triple intersection unless the wording says "exactly those two".

Adding 12 and 9 counts the five students in both groups twice. Subtract the overlap once when finding the union.

Start with the deepest overlap, then fill exactly-two and only-one regions. Keep "at least" separate from "exactly".

Among 100 people, 50 study A, 40 B, 30 C; pairwise overlaps AB=20, AC=15, BC=10 include the 5 studying all three. Union=50+40+30-20-15-10+5=80; none=20. Exactly two=(20-5)+(15-5)+(10-5)=30. Only A=50-20-15+5=20. Summing disjoint regions checks the result.

Pairwise overlaps usually include the triple intersection unless the wording says "exactly those two".

A union A∪B contains elements in either set; an intersection A∩B contains elements in both. The complement Aᶜ contains elements of a stated universal set U that are not in A. Equal sets have the same elements; equal cardinalities alone do not make sets equal. The empty set has zero elements and is a subset of every set.

For finite sets, n(A∪B)=n(A)+n(B)−n(A∩B). “Only A” has n(A)−n(A∩B) elements; “exactly one” has n(A)+n(B)−2n(A∩B). “Neither” has n(U)−n(A∪B) elements. For three sets, n(A∪B∪C)=n(A)+n(B)+n(C)−n(A∩B)−n(B∩C)−n(C∩A)+n(A∩B∩C). Pairwise intersections include the triple overlap unless explicitly labelled “exactly two”. Fill the triple region first, subtract it from pairwise intersections, and then fill the only-one regions.

Of 40 students, 25 drink tea, 20 coffee and 10 both. The union is 25+20−10=35; only tea is 15, only coffee 10, and neither 5. In a three-set diagram with exactly-two regions 3,4,5 and triple region 2, exactly two totals 12 and at least two totals 14.

De Morgan's laws are (A∪B)ᶜ=Aᶜ∩Bᶜ and (A∩B)ᶜ=Aᶜ∪Bᶜ. A finite set with k elements has 2^k subsets, including itself and the empty set.

Do not put intersection on the left side of the two-set union identity. A probability Venn diagram does not imply independence merely because two circles overlap.