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Let a a b and let p a b p is a partion of a determine the equivalence relation r on a for which pis


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1. Let A, B be a m ×n matrix and n× m matrix respectively. Prove or disprove the following statements: (a) ...

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 .
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2.Customers arrive at a soft drink dispensing machine according to a Poisson process with rate λ per hour. Let N(t) ...

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.
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3.Let A and B be two sets, with B being a larger set. Let n be the cardinality of A, ...

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)| ≤ _________
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4.Let a = sin b, 0 < b < pi/2. Find, in terms of b, the solutions of sin 2x = ...

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!
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5.Recall that Euclid showed that in Euclidean geometry, every triangle can be circumscribed by a circle in the Euclidean plane. In ...

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.
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6.Let the Universal Set, S, have 129 elements. A and B are subsets of S. Set A contains 61 elements ...

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?
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7.Let U = {3, 4, 5, 6, 7, 8, 9, 10}, A = {4, 6, 8}, B = ...

. Find the following. (Enter your answers as a comma-separated list.) (A ∪ B)'
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1.AU MAT 120 Systems of Linear Equations and Inequalities Discussion

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