SECTION 4.1 Basics of Set Theory 193
Proof. As you might expect the above can be easily demonstrated
through Venn diagrams (see Exercise 1 below). Here, I’ll give a formal
proof of the first result (viz., that “intersection distributes over union”).
Let x ∈ A∩(B∪C) and so x ∈ A and x ∈ B∪C. From this
we see that either x ∈ A and x ∈ B or that x ∈ A and x ∈ C,
which means, of course, that x ∈ (A∩B)∪(A∩C), proving that
A∩(B∪C)⊆(A∩B)∪(A∩C). Conversely, ifx∈(A∩B)∪(A∩C), then
x∈A∩Borx∈A∩C. In either casex∈A, but alsox∈B∪C, which
means thatx∈A∩(B∪C), proving thatA∩(B∪C)⊆(A∩B)∪(A∩C).
It follows thatA∩(B∪C) = (A∩B)∪(A∩C). The motivated student
will have no difficulty in likewise providing a formal proof of the second
distributive law.
Exercises
- Give Venn diagram proofs of the distributive laws:
A∩(B∪C) = (A∩B)∪(A∩C), and A∪(B∩C) = (A∪B)∩(A∪C).- Show that ifA, B⊆U, thenA−B=A∩B′.
- Use a Venn diagram argument to show that ifA, B, C ⊆U, then
A−(B∪C) = (A−B)∩(A−C) and A−(B∩C) = (A−B)∪(A−C).- Show that ifA, B ⊆U, and ifAandBarefinite subsets, then
|A∪B|=|A|+|B|−|A∩B|.