Sets and Venn diagrams

State the universe before taking a complement

Let ξ = {1,2,3,4,5,6,7,8}, A = {2,4,6,8} and B = {5,6,7,8}. The notation 2∈ A says 2 is an element of A; 3∉ A says it is not. n(A) = 4 counts the elements.

The intersection A∩ B = {6,8} contains elements in both sets. The union A∪ B = {2,4,5,6,7,8} contains those in at least one. Each element occurs once even if it belongs to both.

The complement A' = {1,3,5,7} means the elements of this universal set that are not in A. It does not mean all numbers everywhere that are not in A.

A subset is a set, not a single element

{2,4}⊆ A means every element of that set belongs to A. It is a proper subset, {2,4}⊂ A, because it is not all of A. A⊆ A is true, but A is not a proper subset of itself. The empty set varnothing is a subset of every set and has zero elements.

The statement 2⊆ A confuses an element with a set. Use 2∈ A or {2}⊆ A instead.

Count the overlap once

n(A∪ B) = n(A) + n(B)-n(A∩ B) = 4 + 4-2 = 6. Adding both set sizes counts the overlap twice; subtract one copy. The two elements outside the union complete the eight-element universe.

Check your understanding 1 Place the intersection first

A group has 30 pupils. Eighteen study Art, 16 study Music, and 8 study both. Find the number studying at least one and the number studying neither.

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At least one: 18 + 16 − 8 = 26. Neither: 30 − 26 = 4. The Art-only region is 10 and Music-only is 8. Placing all 18 and 16 outside the overlap would overcount.

Translate the words precisely

“In A but not B” is A∩ B', not the whole of A. “Not in either” is (A∪ B)', not the same as “not in both”, which is (A∩ B)'. Test an element that belongs to only one set to see the difference.