Bridge to Abstract Mathematics: Mathematical Proof and Structures

5.2 CONCLUSIONS INVOLVING V AND -+, BUT NOT 3 165

only if there exist real numbers x and y such that (x, y) E C, but

(x, -Y) 4 c-

We negate the definition of linear independence, given in Example 5,

in a similar manner. The set (v,, v,,... , v,) of vectors is not linearly

independent, and in such a case is said to be linearly dependent, if and

only if there exist n real numbers PI, P2,... , fin, such that Plvl +

B2v2 +... + Bnvn = 0, but not all of the betas equal zero, that is Pj # 0

for some j between 1 and n, inclusive. O

EXAMPLE 1 1 Prove that the set C = ((x, x2) 1 x E R) is symmetric with

respect to the y axis, but not to the x axis.

Solution Note first that an ordered pair (x, y) is on the curve C if and only
if y = x2. Thus a picture of the curve C is simply the familiar parabola
that constitutes the graph of the quadratic function y = f(x) = x2 as il-
lustrated in Figure 5.2. The picture certainly bears out our symmetry
claims, but how are we to prove these claims formally?
To show that C is symmetric with respect to the y axis, let (x, y) be
an arbitrarily chosen element of C; we must show that (-x, y) E C. By
definition of C, (- x, y) E C if and only if y = (- x)~. Now since (x, y) E C,
then y = x2. Since x2 = (- x)~, then y = (- x2), as desired. This com-
pletes the proof of y axis symmetry.
On'the other hand, to show that C is not symmetric with respect to
the x axis, note that (2,4) E C since 4 = 22, but (2, -4) 4 C, since
-4 # 22. We have given a specific counterexample to the statement
(Vx)(Vy)[(x, y) E C -, (x, -y) E C], and that's all there is to it!

Here is another important type of problem on which the preceding dis-

cussion has a bearing. Suppose you are asked to show that a set A is a

proper subset of a set B. By Definition 7, Article 1.1, this means A c B

Figure 5.2 Graph indicates y-axis symmetry, but no

x-axis symmetry.