1.3 ALGEBRAIC PROPERTIES OF SETS 33(A nB)UCisthe
shaded region~'n (B U C) is the
crosshatched regionFigure 1.8 Venn diagram suggesting the falsehood of the conjecture
"A n (B u C) = (A n B) u C, for any three sets A, B, and C."
confirm your results for the sets (A n B) u C ( = {2,3,4,5,9, 10)) and
A n (B u C) ( = (2,3,5)), calculated from Example 2.
Thus the conjecture of the preceding paragraph is false. This illustrates
the danger of believing a general conclusion too readily, based on only a
few examples, and especially, without seeing a proof. It also leaves us with
two problems:
- Are there valid distributive laws of set theory?
- Is there anything to be learned from exploring reasons why the two sets
(A n B) u D and A n (B u D) were equal in Example 3?
We discuss (1) immediately while deferring consideration of (2) until Ex-
ample 4.
We approach the distributive laws through an example from elementary
algebra. The problem is to solve the inequalities x2 - 6x - 7 < 0 and
13x - 11) 2 4 simultaneously. Using elementary algebra, we solve the in-
equalities separately to get (- 1,7), and (- a), f] v [5, a)) respectively, as
their solution sets. The simultaneous solutions are those real numbers that
are common to both solution sets, that is, the elements of the set (- 1,7) n
{(- co, $1 u 15, a))], a set having the form A n (B u C). By graphing this
set along a number line, we obtain the result (- 1, f] u [5,7), gotten geo-
metrically by intersecting (- 1,7) with the intervals (- co, 9 and [5, co)
and taking the union of the resulting sets. Put another way, the solution
set equals ((- 1,7) n (- a), $1) v {(- 1,7) n [5, oo)); that is, it equals
(A n B) u (A n C).
The results of this example suggest the possibility that A n (B u C) =
(A n B) u (A n C) is the distributive law we are seeking. This equation
resembles the distributive law for real numbers, a (b + c) = a b + a. c, if
we substitute intersection for "times" and union for "plus," and so seems
like a plausible candidate on that basis as well.