152 METHODS OF MATHEMATICAL PROOF, PART I Chapter 5
by transitivity. The string should begin with (1 + sin x)/cot2 x, end with
sin x/(csc x - I), and involve the quantity (sin2 x)(l + sin x)/(cos2 x) at
an intermediate step. It is left to you, in Exercise 2(b), to write out such
a presentation of this proof (see the solution to Example 6, for further
guidance).
Here is a form of presentation of the preceding proof that you should
not use.
(1 + sin x)/cot2 x = sin x/(csc x - 1)
(1 + sin x)(tan2 x) = sin x/((l/sin x) - 1)
(1 + sin x)(sin2 x/cos2 x) = sin2 x/(1 - sin x)
(sin2 x)(l + sin x)/(cos2 x) = (sin2 x)(l + sin x)/(l - sin x)(l + sin x)
(sin2 x)(l + sin x)/(cos2 x) = (sin2 x)(l + sin x)/(cos2 x)
In this illustration we have given a series of steps that starts with the
equation to be derived (thus effectively assuming; that which is to be
proved) and ends with a tautology, namely, a statement that a quantity
equals itself. This is a logically incorrect presentation of the proof, even
though all the correct trigonometric relationships are there.Some proofs of equality in set theory can be carried out by a transitivity
argument, using results proved in Article 4.1 by the choose method.
EXAMPLE 5 Prove that X - (Y n 2) = (X - Y) u (X - Z) for any three
subsets X, Y, and Z of a universal set U.
Solution Let X, Y, and Z be arbitrary subsets of U. Then
X-(YnZ)=Xn(YnZ)'
= X n (Y' u 2')
= (X n Y') u (X n 2')
=(X- Y)u(X-Z)
Supply the justification for each of these equations. Each step depends on
a result derived in Article 4.1. 0The next example is also from set theory and has a solution similar in
approach to the solution to Example 4. Due to the complex form both quan-
tities involved, we take the approach of converting each to a common third
quantity.
EXAMPLE 6 Prove that intersection distributes over symmetric difference,
that is, for any three sets A, B, and C in a universal set U, A n (B A C) =
(A n B) A (A n C).