30 SETS Chapter 1
In that example U was not specified, nor was it totally clear from the context
what U should be (R and Z would both have been good guesses). But, by
being familiar with set algebra and aware of the identity A - B' = A n B
of set theory, we can solve the problem with no further information.
Finally, the methods of proof we will study in Chapters 4 through 6, used
initially in this text to prove theorems about the algebra of sets, apply to
writing proofs in all other branches of mathematics.
ELEMENTARY PROPERTIES OF SETS
We begin now to formulate conjectures about general properties of sets.
Using two devices described in Article 1.2, the computational results of
specific examples and the evidence provided by pictures, we will attempt
to discover reasonable candidates for theorems of set theory.EXAMPLE 1 Let U = (l,2,3,... ,9, 10) and A = (1,3, 5,7,9). Compute
A u A', A n A', (A')', A u A, and A n A.solution Since A' = (2,4, 6, 8, 101, then A u A' = (1,2, 3,... ,9, 10) = U,
A n A'= 0, (Af)'= (2,4,6,8, lo)'= (1,3,5,7,9) =A, and Au A =
{1,3,5,7,9) = A n A.You should repeat this example by using other subsets of U, such as
B = {l,2,3) and/or C = (3,4,6,8). Also, construct five Venn diagrams,
corresponding to the five sets A u A', A n A', (A')', A u A, and A n A,
derived from A. Note that each diagram should contain only one circle,
labeled A, inside the rectangle corresponding to U. After doing these ex-
ercises, you will probably agree with Conjecture 1.CONJECTURE 1
Let X be any set with universal set U. Then:
(a) X u X' = U
(b) X n X' = 0
(c) X" = X
(d) XuX=X
(e) XnX=X.Let us caution that statements (a) through (e) do not assume the status
of theorem until we provide a rigorous mathematical proof of each. Even
with verification of specific examples and the evidence provided by pictures,
the possibility exists of an example for which the conjectured statement is
false. Later in this article we comment further on weaknesses of "proof"
by Venn diagram.