Sets: union, intersection and difference

A set is a collection of different things, such as the students in a club. Set operations tell you who is in one club, in both, or in neither.

A lecturer must book rooms for two clubs, and some students belong to both. Counting them twice books too many rooms. This lesson shows how to combine sets and count their members correctly.

Say the chess club has Ana, Ben, Cai and Dee. The robotics club has Cai, Dee, Eli and Fay.

The union is everyone in either club. The intersection is everyone in both. The difference "chess minus robotics" is everyone in chess who is not in robotics. These match or, and, and "and not" from logic.

python3 -c "c = {'Ana', 'Ben', 'Cai', 'Dee'}; r = {'Cai', 'Dee', 'Eli', 'Fay'}; print(sorted(c | r), sorted(c & r), sorted(c - r))"

How many students are in at least one club? Do not add 4 and 4. Cai and Dee are in both lists, so adding counts them twice.

Take the overlap away once: 4 + 4 - 2 = 6. This is called inclusion and exclusion. You include both sets, then exclude the double count.

python3 -c "a = set(range(1, 19)); b = set(range(14, 26)); print(len(a), len(b), len(a & b), len(a | b))"

A list can hold the same value twice and keeps its order. A set cannot and does not. The set {3, 1, 3, 2} is just {1, 2, 3}.

A set A is a subset of B when every element of A is also in B.

python3 -c "print(sorted(set([3, 1, 3, 2, 1])), {1, 2} <= {1, 2, 3})"

chess = [1, 2, 3, 4, 5, 6]
robotics = [5, 6, 7, 8]
both = []
for name in chess:
    if name in robotics:
        both.append(name)
print("in both:", both)
print("in either:", len(chess) + len(robotics) - len(both))
# Add 9 to robotics and press Run again. What changes?