Please Show All Steps. Thanks. Assignment Reasoning 1. (1 pt) On a 8 x 8 chessbo
ID: 2901389 • Letter: P
Question
Please Show All Steps. Thanks.
Assignment Reasoning 1. (1 pt) On a 8 x 8 chessboard, the squares are colored a ternately white and black. Thus there are white squares black squares. Each row/column of the chessboard and has Squares It is thus possible to e this chessboard with dominoes (1 x 2 pieces by laying say 4 dominoes per column. (tile means lay the dominoes, so that they cover the chessboard, no two dom noes overlapping Now suppose we remove two squares from the chessboard from DIAGONALLY opposite corners. Suppose one of the squares we remove is white. Now there are White black squares left squares left and Q: Is it possible to cover the modified chessboard (with the two diagonally opposite corners removed) with dominoes? Why? A. No. Since the total number of remaining squares on the chessboard is odd and every domino covers 2 squares and hence can only be used to tile a region with an even number of squares B. No, Since every time we lay down a domino it cov ers one white square and one black square. Thus since the number of white squares is not equal to the num ber of black squares on the modified chessboard, it is impossible C. Yes. It is possible to tile the modified chessboard by placing dominoes, alternating between horizontal and vertical placements in a suitable way D. Yes. Since there are an equal number of white and black squares remaining on the modified chessboard one can tile the modified chessboard with dominoes each covering one white and one black square Answer(s) submitted. incorrect 2, (1 pt) For n a nonnegative integer, either n 0 mod 3 or n E 1 mod 3 or ns 2 mod 3. Ineach case, fill out the following table with the canonical representatives modulo 3 of the expres sions given n mod 3 ns mod 3 2n mod 3 ns 2n mod 3 From this, we can conclude: 2n mod 3 for all n, we conclude that 3 A. Since n does not necessarily divide ns 2n for all nonnegative integers n B. Since n 2n E 0 mod 3 for all n, we conclude that 3 divides n 2n for any nonnegative integer n Answer(s) submitted. incorrect) 3. (1 pt) Which rule of inference is used in each of the following ar guments? Check the correct answers Kangaroos live in Australia and are marsupials. Therefore kangaroos are marsupials A. Conjuction. B. Hypothetical syllogism C. Addition D. Simplification E. Disjunctive syllogism F. Modus tollens Modus ponens 2. If I go swimming, then stay in the sun too long. If I stay in the sun too long, then I will sunburn. Therefore, if I go swimming, then I will sunburn A. Disjunctive syllogism B. Modus tollens C. Modus ponens Hypothetical syllogism. D. E. Simplification F. Conjunction G. Addition 3. Jerry is a mathematics major and a computer science ma jor. Therefore, Jerry is a mathematics major A. Modus ponens B. Addition C. Disjunctive syllogism D. Modus tollens E. Simplification F. Hypothestical syllogismExplanation / Answer
1. 32 white,32 black, 8, 30 white, 32 black. (If one corner is white (1,1) is white where the first is the row number and second in column number, then (1,8) is black and therefore (8,8) is white again).
B. It is a very famous with a very complex proof.
2. B
0
For proof: if n mod 3 = 0. n is of the form 3k hence n3 and 2n are both divisible by 3.
n mod 3 = 1. n is of form 3k +1. n3= 27k3+1+27k2+3k. 2n = 6k + 2. n3+2n = 27k3+27k2+9k+3.
Hence 1, 2 and 0 respectively.
if n is of form 3k+2. n3= 27k3+6 +2 +18k2+12k. 2n = 6k + 3 +1. n3+2n = 27k3+18k2+18k+12.
Hence 2,1 and 0 respectively.
3.) 1 D 2 C 3 E
12.) B
13.) C
14.) B
15.) A
n mod 3 n3 mod 3 2n mod 3 n3 + 2n mod3 0 0 0 0 1 1 2 0 2 2 10