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Practice Problem: the definition of a set
Does the following collection of signals form a set?
$ \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) = some_fcn(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), eg 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)}.
Answer 2
Write it here.
Answer 3
Write it here.
