Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.1 CONCLUSIONS INVOLVING V, BUT NOT 3 OR +. 153

Solution Before attempting a general proof of an unfamiliar theorem, you
should always test the truth of the statement with at least one example.
You should compute both these quantities for U = Z, A = (2,4,5,7},
B = (4,7,11,15), and C = (7,15,23,26). Drawing relevant pictures, in
this case Venn diagrams, can also be helpful. As to the proof, note first
that
A n (B A C) = A n [(B n C') u (B' n C)]
=(AnBnC')u(AnB1nC)
whereas
(A n B) A (A n C) = [(A n B) n (A n C)'] u [(A n B)' n (A n C)]
= [(A n B) n (A' u C')] u [(A' u B') n (A n C)-J
= [(A n B n A') u (A n B n C')] u [(A' n A n C) u (B' n A n C)]
=[@u(AnBnCf)]u[@u(B'nAnC)]
=(AnBnC')u(AnB'nC)
Again, you should supply justifications for the preceding steps.
Having changed the form of both involved quantities to a common
third quantity, we have essentially solved the problem, but there remains
the matter of how to present the proof. The proof may be presented
correctly in one of two ways. Most conveniently, we may simply append
to the two derivations, given earlier, the statement that both given
quantities equal (A n B n C') u (A n B' n C), and so equal each other.
Or else, more desirable from the point of view of someone with experi-
ence (such as an instructor), the proof might be written in the form of
a proof by transitivity, as in
(A n B) A (A n C) = [(A n B) n (A n C)'] u [(A n B)' n (A n C)]
= [(A n B) n (A' u C')] u [(A' u B') n (A n C)]
= [(A n B n A') u (A n B n C')] u [(A' n A n C) u (B' n A n C)]
=[@u(AnBnC')]u[@u(B'nAnC)]
=(AnBnC')u(AnBf nC)
= A n [(B n C') u (B' n C)]
=An(BAC)

A proof by transitivity may also be appropriate for a theorem asserting

an inequality between numbers or a subset relationship between sets, as in

these examples.

EXAMPLE 7 Suppose it is known that IX + yl s 1x1 + IyI for all real

numbers x and y [see Exercise 8(a)]. Prove that lx - z1 5 Ix - yl +

ly - zl for all real numbers x, y, and z.