Suppose in an experiment you flip a coin twice and observe

Question

1. Draw diagrams of the Cartesian products A x B for the following sets:

2. Suppose in an experiment you flip a coin twice and observe the sequence (H for heads, T for tails, like HT = a sequence where you observe heads then tails).

(a) From the description above, write the collection of outcomes that could occur in this experiment.

(b) Write the Power set corresponding to this collection of outcomes.

(c) The power set is useful for describing events in the experiment, which are qualitatively-describable combinations of outcomes that nay be of interest to you as an observer For example, what subset of the Power wet you wrote corresponds to the event A :=“you flipped at most one head”? What about B := “You flipped tails on the second toss"? Or C := ‘You flipped a coin”

3. Show that llal-IbIl ≤ Ia-bI. (Hint: stan with la|=Ia-b+bl and proceed with the triangle inequality...)

4. Let (a_n) be defined by a_n = n. Does (a_n) have a limit? if so, show that converges to that limit, and if not. Show that it does not converge.

5.  Do the following sequences converge to a limit? If so, what limit? Him: try multiplying and dividing by a term involving n.)

6. Show that the function

is not continuous

7. Suppose that f and g are continuous functions on the interval [a, b). Then show that h := f + g is continuous on [a, b].