| Line 23: | Line 23: | ||
===Answer 1=== | ===Answer 1=== | ||
| − | It depends whether you consider the signals 'x(t) = | + | It depends whether you consider the signals '<math>x(t) = someFcn(t)</math>' 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." | From Wikipedia: "Every element of a set must be unique; no two members may be identical." | ||
| Line 29: | Line 29: | ||
'''(a)''' a set | '''(a)''' a set | ||
| − | '''(b)''' not a set (eg x_1(0) = x_3(0)) | + | '''(b)''' not a set (eg <math>x_1(0) = x_3(0)</math>) |
'''(c)''' not a set (see below) | '''(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), | + | Because none of the above periodic functions are injective (ie multiple distinct inputs (t) may result in same output (x), like <math>x_1(0) = x_1(pi) = 0</math>), <math>\{x_1(t), x_2(t), x_3(t), x_4(t)\}</math> does not comprise a set, nor do <math>\{x_1(t)\}</math>, <math>\{x_2(t)\}</math>, <math>\{x_3(t)\}</math>, or <math>\{x_4(t)\}</math>. |
===Answer 2=== | ===Answer 2=== | ||
Revision as of 14:13, 7 January 2013
Contents
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.
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)\} $.
Answer 2
Write it here.
Answer 3
Write it here.
