Sets: first steps and Venn diagrams

A set is a collection of objects for which we can decide unambiguously whether an object belongs to the collection. The objects are its elements. This lesson starts with essential symbols and reaches the first operations; the Venn diagram lab lets you see precisely which regions each operation selects.

Elements, membership and representations

We write 2 ∈ A when 2 belongs to A and 7 ∉ A when it does not. For example, A={2,4,6} lists its elements; the same collection can be described by a rule: “positive even integers less than 7.” Order and repetition in the list do not matter: {2,4,2}={4,2}. A rule needs a clear domain; “large numbers” does not specify a set without a precise criterion.

The empty set ∅ has no elements. Do not confuse it with {∅}, which has one element—the empty set itself. Thus |∅|=0 and |{∅}|=1. The notation |A| is the number of elements in a finite set A.

Subsets and equality

We write A ⊆ B if every element of A also belongs to B. For example, {2,4} ⊆ {1,2,3,4}. Every set is a subset of itself, and ∅ is a subset of every set. Two sets are equal when they have exactly the same elements: A=B if and only if A ⊆ B and B ⊆ A. Notice the distinction between 2 ∈ A and {2} ⊆ A: the first concerns an element, the second a set.

The universe and first operations

Choose a universe U, the collection of objects under discussion. Let U={1,2,3,4,5,6,7,8,9,10}, A={2,4,6,8,10} and B={3,6,9}. Then:

• Union A∪B={2,3,4,6,8,9,10}: in A or B, including both.
• Intersection A∩B={6}: in A and B simultaneously.
• Difference A∖B={2,4,8,10}: in A but not B.
• Complement U∖A={1,3,5,7,9}: in U but not A.
• Symmetric difference A△B={2,3,4,8,9,10}: in exactly one of the two sets.

The complement changes when U changes. Even when we compare only A and B, the area outside both circles remains part of U. “Or” in a union is inclusive: 6, which belongs to both, is counted only once.

Reading a Venn diagram

The rectangle represents U and each circle represents a set. Each point of the rectangle lies in an atomic region determined by its membership: A only, B only, both, or neither. Three sets have up to 8 membership combinations. Some regions may be empty for the actual sets chosen. In the lab you can switch from two to three sets and see the full operation shaded on the diagram.

To check an operation, imagine any element and ask: “Is it in A? Is it in B?” A∖B requires yes to the first question and no to the second. The complement of A requires no to A, even if the element lies outside every circle.

First properties and a counting rule

We have A∪∅=A, A∩∅=∅, A∪A=A and A∩A=A. Also A∩B ⊆ A ⊆ A∪B. Complements satisfy A∪(U∖A)=U and A∩(U∖A)=∅. Two useful De Morgan identities are U∖(A∪B)=(U∖A)∩(U∖B) and U∖(A∩B)=(U∖A)∪(U∖B). Check them by asking when an arbitrary element belongs to each side.

For finite sets, |A∪B|=|A|+|B|−|A∩B|. Common elements were counted twice, so we subtract one copy. If |A|=12, |B|=9 and |A∩B|=4, the union contains 17 elements.

Twelve exercises with solutions

Try each problem before opening its answer. The first four establish the basics, the next five practise operations and the last three need an extra check.

1. List the multiples of 3 in U={1,…,10}.
   Solution

   {3,6,9}; 12 is outside the universe.

2. If A={2,4}, are 2∈A and {2}⊆A both true?
   Solution

   Yes. The number 2 is an element; {2} is a set whose only element belongs to A.

3. Find |∅| and |{∅}|.
   Solution

   They are 0 and 1. The second set has one element, namely ∅.

4. Check whether {2,4}⊆{1,2,3,4}.
   Solution

   Yes: both 2 and 4 belong to the set on the right.

5. For A={1,3,5}, B={3,4}, find A∪B.
   Solution

   {1,3,4,5}; the shared element 3 appears only once.

6. For the same A and B, find A∩B.
   Solution

   {3}, the only element in both sets.

7. For the same A and B, find A∖B and B∖A.
   Solution

   A∖B={1,5}; B∖A={4}. Difference is not commutative.

8. If U={1,…,6} and A={2,4,6}, find U∖A.
   Solution

   {1,3,5}: exactly the elements of U not in A.

9. If A={1,2,3} and B={3,4}, find A△B.
   Solution

   {1,2,4}. The shared element 3 is excluded.

10. With U={1,…,5}, A={1,3}, B={3,5}, check a De Morgan law.
    Solution

    A∪B={1,3,5}, so U∖(A∪B)={2,4}. Also (U∖A)∩(U∖B)={2,4}.

11. If |A|=12, |B|=9 and |A∩B|=4, find |A∪B|.
    Solution

    12+9−4=17. Subtract the intersection once because it was counted twice.

12. Of 20 students, 12 play chess, 11 paint and 5 do both. How many do at least one activity, and how many do neither?
    Solution

    At least one: 12+11−5=18. Neither: 20−18=2. Check: 18+2=20.

From the lab to advanced problems

In the Venn lab, choose union, intersection, difference, symmetric difference or complement; then colour regions freely to invent your own selection. With three sets, union is A∪B∪C, intersection is A∩B∩C and difference is A∖(B∪C). Symmetric difference A△B△C includes regions belonging to one or all three sets; the complement remains U∖A. Once these first steps are clear, try Sets: 10 very challenging problems with diagrams.