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Algebra › Venn diagrams

Venn diagrams

Two overlapping circles, four regions. The simplest picture of how sets fit together — and the vocabulary for talking about it.

Venn diagram

Tap an operation to shade the region.

ABU (universe)

Showing

A ∩ B

Intersection — in both A AND B.

Example

A = { 1, 2, 3, 4 } · B = { 3, 4, 5, 6 }

= {3, 4}

The setup

A set is just a collection of things — numbers, words, people, anything. Two sets, A and B, drawn as two overlapping circles, split a rectangle (the "universe") into four regions:

  • Only A — in A but not in B.
  • Only B — in B but not in A.
  • Both (the overlap) — in A AND in B.
  • Neither (outside the circles) — not in A and not in B.

The operations

  • Union A ∪ B — everything in A OR B (or both). Read "A union B."
  • Intersection A ∩ B — only things in both A AND B. Read "A intersection B."
  • Difference A − B — in A but not in B. Sometimes written A \ B.
  • Complement Aᶜ or ¬A — everything in the universe that's not in A.

Worked example

Let A = students who play soccer = { Ali, Maya, Sam, Jess }
Let B = students who play basketball = { Sam, Jess, Tom, Nia }

  • A ∩ B (play both) = { Sam, Jess }
  • A ∪ B (play either) = { Ali, Maya, Sam, Jess, Tom, Nia }
  • A − B (play soccer but not basketball) = { Ali, Maya }
  • (A ∪ B)ᶜ (play neither) — whoever's left in the class.

Counting trick

|A ∪ B| = |A| + |B| − |A ∩ B|. Add the two set sizes, then subtract the overlap (otherwise you'd count it twice). For the soccer example: |A ∪ B| = 4 + 4 − 2 = 6. ✓

Three-circle diagrams

Add a third circle and you get 7 regions instead of 4. They're harder to draw cleanly (the three circles must intersect each other pairwise), but useful for problems with three categories.

De Morgan's laws

Two rules that flip complement across operations:

  • (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
  • (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

In words: "not (A or B)" is the same as "not-A and not-B." Draw it on the Venn diagram and you'll see why.

Try it

  1. Let A = { multiples of 2 from 1 to 12 } and B = { multiples of 3 from 1 to 12 }. Find A ∩ B.
  2. Using the same A and B, find A ∪ B.
  3. In a class of 30, 18 take French and 14 take Spanish. 8 take both. How many take neither?
  4. State De Morgan's laws in plain English.

Answers: 1) { 6, 12 } (multiples of 6). 2) { 2, 3, 4, 6, 8, 9, 10, 12 }. 3) Take at least one = 18 + 14 − 8 = 24, so neither = 30 − 24 = 6. 4) "Not in either" = "not in the first AND not in the second"; "not in both" = "not in the first OR not in the second."