Group 4 Project. James Earthman, Gregg Greenleaf, Zach Harvey, Zac Leids

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1 Group 4 Project James Earthman, Gregg Greenleaf, Zach Harvey, Zac Leids

2 2.5 # Give the output signals S and T for the circuit in the right column if the input signals P, Q, and R are as specified. Note that this is not the circuit for a full-adder. a. P = 1, Q = 1, R = 1 b. P = 0, Q = 1, R = 0 c. P = 1, Q = 0, R = 1

3 2.5 #21 Solution a. S = 0, T = 1 b. S = 1, T = 0 c. S = 0, T = 0 A Half-Adder takes two binary digits and adds them together. The output for a half-adder is given in two parts, the carry and the sum. The carry is returned true when both of the inputs are true or 1, this is like carrying a digit in decimal addition. The sum is returned true when one of the inputs is true but not both. Both the carry and the sum are returned false if both inputs are false. This circuit also uses an AND gate. An AND gate only returns a true value when both inputs are true, it returns false in any other case.

Gottfried Leibniz Gottfried Leibniz was the mathematician who invented the Binary number system. Leibniz was a German academic that excelled in areas of mathematics and philosophy.

4 Gottfried Leibniz Gottfried Leibniz was the mathematician who invented the Binary number system. Leibniz was a German academic that excelled in areas of mathematics and philosophy. He is infamously known for inventing Calculus independant of Isaac Newton. Newton later claimed that Leibniz had stolen his work. Although he was an amazing mathematician, he also wrote on politics and law. Binary number theorem is the foundation for modern computing.

5 2.5 # Add and convert the result to decimal notation, to verify that = (2 8 1)10.

6 2.5 #22 Solution = > Decimal = 256 or 2 8 When adding binary numbers, 1 is carried when both digits are 1 this leaves behind a sum of 0. In this problem, the a 1 is carried all the way to the ninth digit leaving behind 0 s is each column. When converting the binary number to decimal when can assign a binary value to each digit via 2 Digit N, then take the sum of those values and calculate our decimal number. For this problem, we see that there is a 1 in the 8th digit giving us a value of 256, since none of the other digits are true our answer is just 256.

7 2.5 #47 Part 1: Convert to decimal notation. Part 2: Convert to decimal notation.

8 2.5 #47 Solution Part 1: = 6 * * * * * 8 0 = Part 2: = 2 * * * * * 8 0 = 8691

9 3.3 #56 Let P(x) and Q(x) be predicates and suppose D is the domain of x. In 55 58, for the statement forms in each pair, determine whether (a) they have the same truth value for every choice of P(x), Q(x), and D, or (b) there is a choice of P(x), Q(x), and D for which they have opposite truth values. x D, (P(x) Q(x)) and ( x D, (P(x)) ( x D, Q(x))

10 3.3 #56 Solution a) No matter what the domain D or the predicates P(x) and Q(x) are, the given statements have the same truth value. If the statement x D, (P(x) Q (x)) is true, then P(x) Q(x) is true for the x chosen in D, which implies that both P(x) and Q(x) are true for the x chosen in D. But then P(x) is true for the x chosen in D, and also Q(x) is true for the x chosen in D. So the statement ( x D, (P(x)) ( x D, Q(x)) is true. Conversely, if the statement ( x D, (P(x)) ( x D, Q(x)) is true, then P(x) is true for the x chosen in D, and also Q(x) is true for the x chosen in D. This implies that both P(x) and Q(x) are true for the x chosen in D, and so P(x) Q(x) is true for the x chosen in D. Hence the statement x D, (P(x) Q(x)) is true.

11 3.3 #58 Let P(x) and Q(x) be predicates and suppose D is the domain of x. In 55 58, for the statement forms in each pair, determine whether (a) they have the same truth value for every choice of P(x), Q(x), and D, or (b) there is a choice of P(x), Q(x), and D for which they have opposite truth values. x D, (P(x) Q(x)), and ( x D, P(x)) ( x D, Q(x))

12 3.3 #58 Solution a) If the statement x D, (P(x) Q(x)) is true, then P(x) Q(x) is true for the x chosen in D, since only P(x) or Q(x) has to be true for the chosen x in D. So the statement ( x D, (P(x)) ( x D, Q(x)) must be true. Conversely, if the statement ( x D, (P(x)) ( x D, Q(x)) is true, then P(x) is true for the x chosen in D, or Q(x) is true for the x chosen in D. This implies that P(x) or Q(x) are true for the x chosen in D, and so P(x) Q(x) is true for the x chosen in D. Hence the statement x D, (P(x) Q(x)) is true.

13 References htm

Advanced Information Storage 02

Advanced Information Storage 02 dvanced Information Storage 02 tsufumi Hirohata Department of Electronics 16:00 10/October/2013 Thursday (V 120) Quick Review over the Last Lecture Von Neumann s model : Memory access : Bit / byte : 1