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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.
You will receive feedback from your instructor and TA directly on this page. Other students are welcome to comment/discuss/point out mistakes/ask questions too!
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
Write it here.
Answer 3
Write it here.
