Let a and b be the two points in the plane that are the nodes of a grid described in

s: (a) (5 marks) (A + B) T = AT + BT . (b) (5 marks) (AB) T = BT AT . (c) (5 marks) AT A is symmetric. (d) (5 marks) AT A = AAT .

hour. Let N(t) be the number of custoer arrivals up to time t, with hour as the unit. There are two types of soft drinks, type A and B, stored in the machine. Suppose that each time a customer deposits money, the machine dispenses one soft drink A with probability p1, or one soft drink B with probability p2. We have p1 + p2 = 1, p1 > 0, p2 > 0. Let X(t) be the number of type A soft drinks dispensed up to time t; and Y (t) be the number of type B soft drinks dispensed up to time t.

be the cardinality of B, so m = |B|. As B is larger, n < m. What can we say about the cardinality of: _________ ≤ |A ∪ B| ≤ _________ _________ ≤ |A \ B| ≤ _________ _________ ≤ |A × B| ≤ _________ _________ ≤ |(A × B) \ (A × A)| ≤ _________ _________ ≤ |(A ∪ B) × (B \ A)| ≤ _________ _________ ≤ |(A ∪ B) × (A \ B)| ≤ _________

So far I just tried to do the obvious and substitute -a for - sin b. Like so; sin 2x = - sin b which then becomes; sin 2x = sin -b and then I can remove the sines and rearrange for x, which gives me this. 2x = -b x = -b/2 However, if I were to look at the domain of b. Any value within the domain will give a value that doesn't fit the domain of x. Any suggestions would be great. Thank you!

e in the Euclidean plane. In this exercise, we will show this is not true in the Poincar´e plane. As usual, let Γ be the unit circle in R2 , and let O be the origin. Let A = (3/4, 0) and B = (0, 3/4). Consider the P-triangle △OAB. Let γ be the E-circle through O, A, and B. Show that γ contains points outside of Γ and is therefore not a P-circle. Explain why this means that the P-triangle △OAB cannot have a circumscribed circle in the Poincar´e plane.

d Set B contains 39 elements. If the total number of elements in either A or B is 84, how many elements are in A but not in B? How do I do this?

. Find the following. (Enter your answers as a comma-separated list.) (A ∪ B)'