SECTION 4.1 Basics of Set Theory 191
I’m sure that you’re reasonably comfortable with these notions. Two
other important construction are the difference and complement,
respectively:
A−B = {u∈U|u∈Abutu6∈B}, andA′ = {u∈U|u6∈A} = U−A.Relationships and operations regarding subsets are often symbolically
represented through the familiarVenn diagram. For example, in the
Venn diagram below, the student should have no difficulty in coloring
in any one of the subsetsA∪B, A∩B, A−B, B−A, A′(or any others
that might come to mind!)
Venn diagrams can be useful in identifying properties of the above
operations. One very typical example of such relationships and their
Venn diagram proofs are theDe Morgan Laws: for subsetsAandB
of a universal setU, one has
(A∪B)′=A′∩B′ and (A∩B)′=A′∪B′.You can convince yourself of these facts by coloring in the Venn
diagrams: