Discrete math -content– 1. Consider the set 𝑆={(𝑥1,𝑥2,…,𝑥𝑛)?ℤ𝑛|𝑥𝑖=0 𝑎𝑛𝑑 𝑥1+𝑥2+⋯+𝑥𝑛=𝑘𝑚}. Throughout the problem assume that 𝑘,𝑚?ℤ and 𝑘=2,𝑚>0. a. Assume 𝑘=2. Describe a method of counting the number of elements in 𝑆 whose entries are all even. (that is, explain how to count the number of ordered non-negative solutions to 𝑥1+𝑥2+⋯+𝑥𝑛=2𝑚 in which all the 𝑥𝑖 are even) b. Assume that 𝑚=𝑛=3. If you randomly select an element of S, what is the probability that its coordinates are all equivalent 𝑚𝑜𝑑 𝑘? Hint: there are two distinct cases to consider – 𝑘|𝑛 and 𝑘∤𝑛. 2. Consider the set of vertices, 𝑉={1,…,𝑛}. The set 𝐺𝑛={(𝑖,𝑗)?𝑉×𝑉|𝑖<𝑗} represents a graph. (recall the definition from class that an ordered pair represents an edge between 𝑖 and 𝑗) a. For which values of 𝑛 does 𝐺𝑛 contain an Eulerian circuit? b. How many distinct Hamiltonian circuits does 𝐺𝑛 contain? Hint: it might be more intuitive to count Hamiltonian paths and then investigate how many circuits each of these paths can extend to. 3. Let 𝑃={𝑝?â„• | 𝑝>2 𝑎𝑛𝑑 𝑖𝑓 𝑝=𝑎𝑏 𝑡ℎ𝑒𝑛 𝑎=1 𝑜𝑟 𝑏=1}. a. Find the equivalence classes of 𝑃 under congruence 𝑚𝑜𝑑 4. b. If 𝑝?𝑃 and ?𝑥,𝑦?ℤ such that 𝑝=𝑥2+𝑦2, which of the equivalence classes listed in part a) is 𝑝 an element of? Hint: recall that if 𝑎=𝑏 𝑚𝑜𝑑 𝑛 and 𝑐=𝑑 𝑚𝑜𝑑 𝑛 then 𝑎+𝑐=𝑏+𝑑 𝑚𝑜𝑑 𝑛 and also 𝑎𝑐=𝑏𝑑 𝑚𝑜𝑑 𝑛. PLACE THIS ORDER OR A SIMILAR ORDER WITH US TODAY AND GET A GOOD DISCOUNT

2016-09-21