Question 1 Coursology 4 pts Consider your student ID as a sequence of eight digits. For example, if your student ID is the number 21238586, then it represents the sequence D = (2, 1, 2, 3, 8, 5, 8, 6). The sequence can be used to define four two-digit integers as follows, n1 = did2, n2 = d3d4, n3 = dad6, n4 = drdg The di, 1 pe {3, 5, 7} | (m -n). Or m is related to n by R if, and only if, at least one of the numbers 3, 5, or 7 divides m - n. Record the relation R as a set of 2-tuples. Consider one related pair in the example set S, [(86, 21 E S) and (86 - 21 = 65) and (5 | 65) - (86, 21) ( R]. For the example, R = {. .., (86, 21) , . . .}, where the ellipses stand for all the other tuples in the example relation. Your answer must show all, and only, 2-tuples in R created by your student ID and its derived set S.Question 7 REJTEILIY 4 pts The relation R is defined on Z as follows: Vm,neZ, mRn Jdke Z | (mmn)=2k [(mn)isaneven number] Prove that the relation R is an equivalence relation. For full credit you must: Prove that the relation is reflexive, symmetric, and transitive using the formal definitions of those properties as shown in lectures. Give your proof line-by-line, with each line a statement with its justification. Show explicit, formal start and end statements for the overall proof and for the proof case for each property. You can use the Canvas math editor or write your math statements in English. For example, the universal statement that is to be proved was written in the Canvas math editor. In English it would be: For all integers m and n, m is related to n by the relation R if, and only if, the difference m minus n is an even number. Question 8 Coursology 4 pts Consider your eight-digit student ID as a set of single-digit integers. For example, if your student ID is the number 01238586, then it represents the set S = {0, 1, 2, 3, 5, 6, 8}. Now consider your student ID as a sequence of eight digits. For example, if your student ID is the number 01238586, then it represents the sequence D = (0, 1, 2, 3, 8, 5, 8, 6). The sequence can be used to define a relation r:S -> S by creating the elements of r as follows, r = {(d1, d2) , (d3, d4) , (ds, do) , (d7, d8) } The di, 1