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Worksheet 1 : Components of a Mathematical Model, Set Theory The solutions are available in my office. However, to get them, you need to show me the answers to the questions first. 1. In a class of 120 students numbered 1 to 120, all even numbered students opt for Physics, numbers that are divisible by 5 opt for Chemistry and those whose numbers are divisible by 7 opt for Math. How many opt for none of the three subjects? Hint: Use the following formula to calculate the answer: 120 – PUCUM, where PUCUM = P+C+M - (P n C + C n M + M n P) + (P n C n M) P = no of students who opted for Physics C= no of students who opted for Chemistry M= no of students who opted for Maths 2. Shade the following in a Venn Diagram. Assume all sets interact with each other (i.e. when drawing A, B and C, they will overlap with each other, whereas when drawing only A and B (as in Question 7ii), they will overlap each other: (i) (ii) (iii) (iv) (A – B) U C (AnB) ′ where ′ indicates complement of a set (A′∩B) ∩ (BUC) ′ Suppose we have 3 sets: Set A, Set B and Set C. Assume Sets A and B do not have any elements common between themselves. However, both the sets have all common elements with Set C, which is a much bigger set than A & B. Draw a Venn diagram carefully showing the 3 sets and shade the following region: (AUB) c C (A union B is a subset of C) 4. Write down the following sets either by enumeration or by description. (i) The set of all integers greater than –5 but less than 5 (ii)The set of all prime numbers from 0 to 25 (a prime number is any number that is greater than 1 and is divisible by itself and the number 1 only) (iii)The set of all real numbers greater than 0 5. Let S = {1,2,3}, T = {3,4,5}, V = {3,2,1} and U = {1,2,3,4,5}. Which of the following statements are correct? If a statement is incorrect, correct it. (a) S = T (b) S = V (c) 3 Є S (d) 4 Є V (e) S c V (f) T c S (g) V c T (V is not a subset of T) (h) SUT≠ U (i) SnT = U (j) VnT = Ø (k) S U V=S (l) U – S = T (m) V′ =T (n) U – S = U – V (o) S U V U T=U (where ′ indicates complement) 1 6. Suppose there are 820 students in College S. Out of these 820 students, 400 study Mathematics, 250 study Physics and 150 study chemistry. In addition, you are told that 550 students study either maths or physics or both, 370 students study either physics or chemistry or both, 480 students study either Maths or Chemistry or both. Also, you are told that 610 students take Maths or Chemistry or Physics or all. Assuming M=students studying Mathematics, P=students studying Physics and C=students studying Chemistry, draw a Venn diagram and show values of the following: M, P, C, MnP, MnC, PnC and MnPnC. NB: Show the calculations separately. 7. Suppose we have 5 friends in NSU. Lets name them: Sunny (S), Johnny (J), Bonny (B), Canny (C) and Danny (D). Assume S, B and D are members of Debating Club, S, C & D are members of the YEF Club and S, J, B and C are members of the Earth Club. Suppose DC represents debate club, YC represents YEF club and EC represents Earth Club. Using the information above, draw a Venn diagram and determine which member/members have joined ONLY the earth club? 8. Which of the following equations are functions and why? (v) (vi) (vii) (viii) y = -2x +7 y2 = x y = x2 x2 + y2 =64 9(i) Roughly plot the demand function of the original good given as Qd = -4P + 0.01Y – 5Pr + 10T When Y=8000, Pr = 8 and T=4 (ii) What type of good is the related good with respect to the original good? 10. Plot the following functions: (i) y = 5 +2x + x2 (ii) y = (x – 8)2 2 11. Given A = { 1,2,3,4,5} and B = {5}, find the Cartesian Product A x B. 12. The total revenue, which is defined as Tr = PxQ, of a firm per day is a function of its daily sales Q. Assume that the firm’s output capacity is Q = 10 units per day. What are the domain and range if the total revenue function is given as TR = 5Q – 1/2Q2? 13. Solve for x : a) 2x-5 = 32 b) ln (5x − 1) = ln (2x + 8). c) 5x + 1 = 625 14. Change the following logarithms to their equivalent exponential forms: a) Log5 125 = 3 b) Log16 2 = ¼ c) Loga y = 6x d) Log2 y = 7x e) Log7 1 7 = −1 15 . Change the following exponentials to logarithmic forms: a) 32= 25 3 b) 27 = 92 c) 1 9 = 3−2 16 a) Simplify loga 56 + Loga x b) Expand ln(x5√y) 5 c) Expand 3 √ √ 3