(a) Find ​P(T<1.321) when v=22. ​(b) Find ​P(T>2.069​) when v=23. ​(c) Find ​P(−2.145<T<2.997​) when v=14. ​(d) Find ​P(T>−2.998) when v=7.

Specify sample spaces for the random experiments (a)-(e) and give mathematical descriptions of the corresponding events. (a) Experiment: A coin is tossed three times. Event: The result of the second toss is “heads”. (b) Experiment: Three indistinguishable coins are tossed at the same time. Event: At most two of the coins show “heads”. (c) Experiment: A die is rolled until each number has appeared at least once. The outcome is the number of rolls needed. Event: Less than 15 rolls are needed. (d) Experiment: n devices labeled with 1, . . . , n are inspected. It is of interest which devices are working and which are not. Event: The first three devices are defective. (e) Experiment: n devices are inspected. Interest lies in the number of defective devices. Event: Exactly three devices are defective.

145&146&147. The potential of a silver electrode is measured relative to an Ag-AgCl electrode for the titration of 100.0 mL of 0.100 M Cl with 0.100 M Ag+. What is the potential after 75.00 mL of titrant is added. Eo = 0.799 V for Ag+, E = 0.197 V for the Ag-AgCl electrode and Ksp = 1.8 × 10−10.

A) 0.493 V

B) 1.070 V

C) 0.521 V

Let V be the set of all ordered pairs of real numbers. Consider the following addition and scalar multiplication operations V. Let u = (u1, u2) and v = (v1, v2).

Show that V is not a vector space.

• u ⊕ v = (u1 + v1 + 1, u2 + v2 + 1 )

• ku = (ku1 + k − 1, ku2 + k − 1)

1)Show that the zero vector is 0 = (−1, −1).

2)Find the additive inverse −u for u = (u1, u2). Note: is not (−u1, −u2), so don’t write that.

3)Show that V is not a vector space.

I have a collection of baseball cards, some from the 1984 season and some from other seasons. Considering only my valuable baseball cards, 60% of them are from 1984. Overall, 3% of my baseball cards are both valuable and from the 1984 season, and 38% of my cards are not valuable and not from the 1984 season. Fill out the table below

p(1984 and V) =

p(not 1984 and V) =

p(V) =

p(1984 and not V) =

p(not 1984 and not V) =

p(not V) =

p(1984) =

p(not 1984) =

GVC_E(V,E)

1. C =Φ

2. while ( E≠ Φ)

3. do

4. { select an edge (u,v)∈E;

5. C = C∪{u}∪{v};

6. delete u and v from V and edges with u or v as an endpoint from E;

7. }

8. for each u∈C

9. { if C\{u} is a valid cover;

10. C = C\{u};

11. }

How can I complete the pseudo code on line 4 and line 8 by adding a heuristic specifying how to select a node

a.)  Let T be a binary tree with n nodes. Define the lowest common ancestor (LCA) between two nodes v and w as the lowest node in T that has both v and w as descendants. Given two nodes v and w, write an efficient algorithm, LCA(v, w), for finding the LCA of v and w. Note: A node is a descendant of itself and v.depth gives a depth of a node v.

b.) What is the running time of your algorithm? Give the asymptotic tight bound (Q) of its running time in terms of n and justify your answer.

Let f : V mapped to W be a continuous function between two topological spaces V and W, so that (by definition) the preimage under f of every open set in W is open in V : Y is open in W implies f^−1(Y ) = {x in V | f(x) in Y } is open in V. Prove that the preimage under f of every closed set in W is closed in V . Feel free to take V = W = R^n to simplify things. Hint: show that the “preimage of” operation plays nice with set-complements, and then use the fact that every closed set is the complement of some open set. Note that R^n is both open and closed as a subset of itself.

For each of the following statements, determine whether the statement is true or false. If you say the statement is true, explain why and if you say it is false, give an example to illustrate.

(a) If {u, v} is a linearly independent set in a vector space V, then the set {2u + 3v, u + v} is also a linear set independent of V.

(b) Let A and B be two square matrices of the same format. Then det (A + B) = det (A) + det (B).

(c) It is possible to find a non-zero square matrix A such that A^2 = 0.

(d) Let V be a vector space. If {v1, v2,. . . , vn} (with n ≥ 1) is a base of V and if {w1, w2,. . . , wm} is a generator system of V then n ≤ m.