The concept of a set, operations on sets, the complement of a set
A set is one of the basic notions accepted without definition: it is a collection of things viewed as one whole, and the things in it are called its elements. “x belongs to A” is written x ∈ A and “x does not belong to A” is written x ∉ A. A set with a finite number of elements is finite, otherwise it is infinite; the number of elements of a finite set A is n(A), and the set with no elements is the empty set ∅ (n(∅) = 0). If every element of A also belongs to B, then A is a subset of B: A ⊆ B. The union A ∪ B consists of the elements lying in at least one of the sets, and the intersection A ∩ B of the elements lying in both; sets with no common element are disjoint. When a universal set U is given, the complement A′ contains all elements of U that are not in A: A ∩ A′ = ∅, A ∪ A′ = U, n(A) + n(A′) = n(U). For finite sets n(A ∪ B) = n(A) + n(B) − n(A ∩ B), because the common elements are counted twice in the sum.
Make two overlapping circles on the floor (with rope or chalk): “learn English” and “play chess”. Students stand where they belong; those in the overlap belong to both. Count and check n(A ∪ B) = n(A) + n(B) − n(A ∩ B).