Revision as of 08:43, 8 January 2013

Practice Problem: the definition of a set

Does the following collection of signals form a set? (Revised)

$ \begin{align} x_1(t) &= \sin t \\ x_2(t) &= \cos t \\ x_3 (t) &= \sin \frac{t}{2} \\ x_4(t) & = -\sin \left(t-\frac{\pi}{2} \right) \end{align} $

Justify your answer.

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Answer 1

It depends whether you consider the signals '$ x(t) = someFcn(t) $' as (a) character strings, (b) input/output pairs (t,x), or (c) the outputs (x) for all valid inputs (t). I assume that case (c) was intended for consideration here.

From Wikipedia: "Every element of a set must be unique; no two members may be identical."

(a) a set

(b) not a set (eg $ x_1(0) = x_3(0) $)

(c) not a set (see below)

Because none of the above periodic functions are injective (ie multiple distinct inputs (t) may result in same output (x), like $ x_1(0) = x_1(pi) = 0 $), $ \{x_1(t), x_2(t), x_3(t), x_4(t)\} $ does not comprise a set, nor do $ \{x_1(t)\} $, $ \{x_2(t)\} $, $ \{x_3(t)\} $, or $ \{x_4(t)\} $.

Instructor's comment: Actually, it's neither a) nor b) nor c). The question is whether the "signals" themselves are all distinct. Good thinking process though, and very well articulated. Keep up the good work! Anybody else wants to venture a guess? -pm

Answer 2

These 4 elements do not form a set. Again the definition of a set is that it must contain unique elements. Signal 4 can be reduced to signal 2; therefore, these two signals are the same and thus not a set. The math is quite simple so I won't work it out, but if someone doesn't understand I would be happy to. Instructor's comment: Very good! Clearly explained, all the important elements are there. Answer is correct. -pm

Answer 3

Write it here.

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