1. Is it a function If so, onto one to one bijection explanation.pdf

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1. Is it a function? If so, onto? one to one? bijection? explanation please f : R R where f(x) = 1/(x - 3) 2. Is it a function? If so onto? one to one? bijection? Explanation please f : Z^2 Z^3 where f(x,y) = (x, y, x + y) Solution 1. If f(x) = f(y), then 1/(x-3) = 1/(y-3) or, y-3 = x -3 so that x = y. Thus, the function f: R R where f(x) = 1/(x - 3) is one-to-one. Further, let y = 1/(x-3) . Then x-3 = 1/y or, x = 3+1/y. On interchanging x and y, we get y = 3+1/x. Thus x R is the image of (3+1/x ) R. Thus every element x of R has a pre-image under f in R. Hence f is onto. Therefore f is a bijection. 2. If f(x1,y1) = f(x2,y2), then (x1, y1, x1 +y1) = (x2, y2, x2 +y2) so that x1 = x2 and y1 = y2 . Hence (x1, y1) = (x2, y2). Hence f : Z2 Z3 defined by f(x , y) = (x, y, x + y) is one-to-one. Further, let (a, b ,c) be an arbitrary element of Z3 such that c a +b. Then (a, b, c) does not have a pre-image under f in Z2 . Hence f is not onto. Therefore, f is also not a bijection..

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Below are the transactions and adjustments that occurred during the first year of operations at Kissick Co a. Issued 190,000 shares of $5-par-value common stock for $950,000 in cash. b. Borrowed $500,000 from Oglesby National Bank and signed a 10% note due in three years. c. Incurred and paid $380,000 in salaries for the year. d. Purchased $730,000 of merchandise inventory on account during the year. e. Sold inventory costing $590,000 for a total of $910,000, all on credit. f. Paid rent of $220,000 on the sales facilities during the first 11 months of the year. g. Purchased $180,000 of store equipment, paying $55,000 in cash and agreeing to pay the difference within 90 days. h. Paid the entire $125,000 owed for store equipment and $620,000 of the amount due to suppliers for credit purchases previously recorded. i. Incurred and paid utilities expense of $35,000 during the year. j. Collected $825,000 in cash from customers during the year for credit sales previously recorded. k. At year-end, accrued $50,000 of interest on the note due to Oglesby National Bank. l. At year-end, accrued $20,000 of past-due December rent on the sales facilities. Required a. Prepare an income statement (ignoring income taxes) for Kissick Co.\'s first year of operations and a balance sheet as of the end of the year. account affected by the transactions.) KISSICK CO Income Statement Solution Answers ASSETS LIABILITIES STOCKHOLDER EQUITY INCOME STATEMENT (Retained earnings) Transaction Cash Accounts receivables Merchandise Inventory Equipment Notes Payable Accounts Payable Common Stock Net Income Revenue Expenses a $ 950,000.00 $ 950,000.00 $ - b $ 500,000.00 $ 500,000.00 $ - c $ (380,000.00) $ (380,000.00) $ 380,000.00 d $ 730,000.00 $ 730,000.00 $ - e $ 910,000.00 $ 910,000.00 $ 910,000.00 $ (590,000.00) $ (590,000.00) $ 590,000.00 f $ (220,000.00) $ (220,000.00) $ 220,000.00 g $ (55,000.00) $ 55,000.00 $ - $ 125,000.00 $ 125,000.00 $ - h $ (745,000.00) $ (745,000.00) $ - i $ (41,000.00) $ (41,000.00) $ 41,000.00 j $ 825,000.00 $ (825,000.00) $ - k $ 50,000.00 $ (50,000.00) $ 50,000.00 l $ 20,000.00 $ (20,000.00) $ 20,000.00 Total $ 834,000.00 $ 85,000.00 $ 140,000.00 $ 180,000.00 $ 550,000.00 $ 130,000.00 $ 950,000.00 $ (391,000.00) $ 910,000.00 $ 1,301,000.00 Transaction no. e Sales Revenues $ 910,000.00 e Cost of Goods Sold $ 590,000.00 Gross Profits $ 320,000.00 Operating Expenses: c Salaries expense $ 380,000.00 f + l Rent Expenses $ 240,000.00 i Utilities expense $ 41,000.00 Total operating expenses $ 661,000.00 Operating Income $ (341,000.00) Other revenues & expenses j Interest expense $ 50,000.00 Net Income (Loss) $ (391,000.00) ASSETS Current Assets: Cash $ 834,000.00 Accounts receivables $ 85,000.00 Merchandise Inventory $ 140,000.00 Total current assets $ 1,059,000.00 Equipment $ 180,000.00 Total Assets $ 1,239,000.00 LIABILITIES Accounts Payable $ 110,000.00 Interest Payable $ 50,000.00 Rent (Expense) Payable $ 20,000.00 Total current Liabilities $ 180,0.

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Microscopes and telescopes both consist of two converging lenses contained in a tube. What is the difference in set-up of the two lenses between a microscope and a telescope? The rearview mirror on a truck warns the user that objects may be closer than they appear. What kind of mirror is being used, and why was that type selected? Why does a clear stream always appear to be shallower than it actually is? Discuss the type of aberration involved in each of the following situations. (i) The edges of the image appear reddish. (ii) The image cannot be clearly focused. A baby fish has his eyesight tested and is found to be myopic. His father wants to make a set of glasses to correct for this problem. Since fish live under water, the father fish will make the glasses out of a very thin clear plastic bag which is filled with air (n = 1.000). What type of lens does the fish need to make? What shape should the lens be? What is the wavelength of yellow light? Solution In refracting telescopes, there are typically two convex lenses. One lens acts as the objective lens: this lens gathers light from faraway objects and forms a real, inverted image of the object at its focal point. A second lens, called the eyepiece, is positioned such that the image formed by the objective lens is at its focal point. When an observer looks through the eyepiece with a relaxed eye, they are able to see an object of the image, formed at infinity. Microscopes are used to look at magnified images of small objects. A simple microscope (a “magnifying glass”) consists of a single convex lens. The lens is held close to the object so that the object is between the lens and its focal point. When viewed from the other side of the lens, a magnified, virtual, upright image is seen. The compound microscope is the most common type of microscope used in laboratories. With these microscopes, an objective lens is used to create an inverted, real image of the object. Using the eyepiece, the image is magnified. In this sense, its operating principles are similar to that of a refracting telescope. The rear view mirror is a convex mirror which forms a virtual and erect image of the object which is smaller in length than the object. Because the image formed is smaller in length, it appears to be farther away and hence, the warning written on it. This is because of the phenomenon of refraction of light as it crosses the medium of water(refractive index greater than that of air). As the light ray travels from water(denser medium) to air(rarer medium), the light ray bends away from the normal and the image of the bottom of the stream seems to be raised and hence, the stream appears to be shallower. a) In optics, chromatic aberration (CA, also called chromatic distortion, and spherochromatism) is an effect resulting from dispersion in which there is a failure of a lens to focus all colors to the same convergence point. It occurs because lenses have different refractive indices for differentwavelength.

Microscopes and telescopes both consist of two converging lenses cont.pdf

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