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3. (a) Fill in True if the following equation is correct and False otherwise. 1 · 10^2 + 2 · 10^1 + 3 · 10^0 = 1 · 2^6 + 1 · 2^5 + 1 · 2^4 + 1 · 2^3 + 0 · 2^2 + 1 · 2^1 + 1 · 2^0 ○ True ○ False (b) In our standard base 10 number system we represent the number given in part (a) as 123. Which of the Left-Hand-Side or Right-Hand-Side of the above equality gives this representation? ○ Left-Hand-Side ○ Right-Hand-Side (c) In the base 2 (binary) number system we represent the number given in part (a) as 1111011. Which of the Left-Hand-Side or Right-Hand-Side of the above equality gives this representation? ○ Left-Hand-Side ○ Right-Hand-Side 4. In this problem we learn a method for encoding each of the subsets of the set {a, b, c} with a binary number. Fill in each of the blank cells in the table below. Subsets of {a, b, c} | Binary representation | Base 10 representation ∅ | 000 | 0 {a} | 100 | 4 | 010 | 2 {c} | | 1 {a, b} | | 6 {b, c} | 011 | 3 | 101 | 5 {a, b, c} | | 7 5. We can think of the table in problem 4 as describing a function. The domain of this function is the set of all subsets of {a, b, c}, otherwise known as the powerset, denoted P({a, b, c}). The range is all of the natural numbers, written in base 10, from 0 to 7. (a) Consider f⁻¹ : {0, 1, 2, 3, 4, 5, 6, 7} → P({a, b, c}) as illustrated in the table above. Fill in the circle for the correct answer. ○ f⁻¹ is a function. ○ f⁻¹ is not a function. (b) Consider f : P({a, b, c}) → {0, 1, 2, 3, 4, 5, 6, 7} as illustrated in the table above. Fill in the circle for the correct answer. ○ f is a bijection. ○ f is not a bijection.
Adi S.
Bit transmission errors between computers sometimes occur, where one computer sends a 0 but the other computer receives a 1 (or vice versa). Because of this, the computer sending a message repeats each bit three times, so a 0 is sent as 000 and a 1 as $111 .$ The receiving computer "decodes" each triplet by majority rule: whichever number, 0 or $1,$ appears more often in a triplet is declared to be the intended bit. For example, both 000 and 100 are decoded as $0,$ while 101 and 011 are decoded as $1 .$ Suppose that 6$\%$ of bits are switched $(0$ to $1,$ or 1 to 0$)$ during transmission between two particular computers, and that these errors occur independently during transmission. (a) Find the probability that a triplet is decoded incorrectly by the receiving computer. (b) Using your answer to part (a), explain how using triplets reduces communication errors. (c) How does your answer to part (a) change if each bit is repeated five times (instead of three)? (d) Imagine a 25 kilobit message (i.e., one requiring $25,000$ bits to send). What is the expected number of errors if there is no bit repetition implemented? If each bit is repeated three times?
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SECTION 2-2 Binary Numbers 5. Convert the following binary numbers to decimal: (a) 11 (b) 100 (c) 111 (d) 1000 (e) 1001 (f) 1100 (g) 1011 (h) 1111 6. Convert the following binary numbers to decimal: (a) 1110 (b) 1010 (c) 11100 (d) 10000 (e) 10101 (f) 11101 (g) 10111 (h) 11111 7. Convert each binary number to decimal: (a) 110011.11 (b) 101010.01 (c) 1000001.111 (d) 1111000.101 (e) 1011100.10101 (f) 1110001.0001 (g) 1011010.1010 (h) 1111111.11111 8. What is the highest decimal number that can be represented by each of the following numbers of binary digits (bits)? (a) two (b) three (c) four (d) five (e) six (f) seven (g) eight (h) nine (i) ten (j) eleven 9. How many bits are required to represent the following decimal numbers? (a) 17 (b) 35 (c) 49 (d) 68 (e) 81 (f) 114 (g) 132 (h) 205 10. Generate the binary sequence for each decimal sequence: (a) 0 through 7 (b) 8 through 15 (c) 16 through 31 (d) 32 through 63 (e) 64 through 75 SECTION 2-3 Decimal-to-Binary Conversion 11. Convert each decimal number to binary by using the sum-of-weights method: (a) 10 (b) 17 (c) 24 (d) 48 (e) 61 (f) 93 (g) 125 (h) 186 12. Convert each decimal fraction to binary using the sum-of-weights method: (a) 0.32 (b) 0.246 (c) 0.0981 13. Convert each decimal number to binary using repeated division by 2: (a) 15 (b) 21 (c) 28 (d) 34 (e) 40 (f) 59 (g) 65 (h) 73 14. Convert each decimal fraction to binary using repeated multiplication by 2: (a) 0.98 (b) 0.347 (c) 0.9028 SECTION 2-4 Binary Arithmetic 15. Add the binary numbers: (a) 11 + 01 (b) 10 + 10 (c) 101 + 11 (d) 111 + 110 (e) 1001 + 101 (f) 1101 + 1011 16. Use direct subtraction on the following binary numbers: (a) 11 - 1 (b) 101 - 100 (c) 110 - 101 (d) 1110 - 11 (e) 1100 - 1001 (f) 11010 - 10111 17. Perform the following binary multiplications: (a) 11 x 11 (b) 100 x 10 (c) 111 x 101 (d) 1001 x 110 (e) 1101 x 1101 (f) 1110 x 1101 18. Divide the binary numbers as indicated: (a) 100 $div$ 10 (b) 1001 $div$ 11 (c) 1100 $div$ 100 SECTION 2-5 1's and 2's Complements of Binary Numbers 19. Determine the 1's complement of each binary number: (a) 101 (b) 110 (c) 1010 (d) 11010111 (e) 1110101 (f) 00001 20. Determine the 2's complement of each binary number using either method: (a) 10 (b) 111 (c) 1001 (d) 1101 (e) 11100 (f) 10011 (g) 10110000 (h) 00111101 SECTION 2-6 Signed Numbers 21. Express each decimal number in binary as an 8-bit sign-magnitude number: (a) +29 (b) -85 (c) +100 (d) -123 22. Express each decimal number as an 8-bit number in the 1's complement form: (a) -34 (b) +57 (c) -99 (d) +115 23. Express each decimal number as an 8-bit number in the 2's complement form: (a) +12 (b) -68 (c) +101 (d) -125
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