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Practice Problemon set operations
Consider the following sets:
$ \begin{align} S_1 &= \left\{ \sin (t), \cos (t)\right\}, \\ S_2 & = \left\{ \sin (\frac{t}{2}), \sin (t+\frac{\pi}{2})\right\}. \\ \end{align} $
Write $ S_1 \cup S_2 $ explicitely. Is $ S_1 \cup S_2 $ a set?
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
No, because a set must have unique elements; sin(t+pi/2) is basically cos(t). The union of both sets is a set with elements from both S1 and S2. S1 U S2 = {sin(t),cos(t),sin(t/2)}
Answer 2
$ S_1 \cup S_2 = \left\{ \sin (t),\sin (\frac{t}{2}), \cos (t)\right\} $
$ S_1 \cup S_2 $ is a set because the union of two sets is the set of all distinct elements from those two sets. In this case because $ \sin (t+\frac{\pi}{2}) $ and cos(t) are part of the same equivalence class, we only need to include one of these elements in our union set.
Instructor's suggestion: Can anyone illustrate the answer using a Venn diagram? -pm
Answer 3
S1 is sub set of S2. In venn diagram, omega which is { real positvie numbers between [-1,1]} will be entire domain. S1 will be included in S2. omega[S2[S1[]]]. I am not sure how to write mathmatical expression in this page and venn diagram.
- YOu would have to draw the Venn diagrams using a drawing program, and then post the jpg file (or image in some other format) using the syntax [[Image:Example.jpg]] . For the math, just type the equation using the syntax <math>Insert formula here</math>. To use special characters in your equation, use latex code. There is a short introduction to posting equations on Rhea on this page. -pm
Instructor's suggestion: Everybody please comment on the above answers! You can also ask questions, or suggest alternative solutions. -pm
Answer 4
Write it here.
I guess the answer is yes, because as the result, the union set has three elements: sin(t), sin(t/2), cos(t). All the elements of the union set is distinct now, which is different from the first practice problem's case.
