At the imaginary university e/UT there are 101 students in their first year at M&CS. In their first year the students need to take k courses. To help them the students are registered for tutor groups. The following conditions hold: o Every student is involved in each course. o Each student is involved in one or more tutor groups. o Each tutor group is involved in one or more courses. o Every student in a course is involved in exactly two tutor groups. o Each pair of students is involved in exactly one tutor group. What is k? (Prove your answer.) (Hint: count the total number of registrations for tutor groups.)

1. Consider the sentence 3xVy(A(x,y) → (B(y) ^ C (@))) (a) Build a model, using the natural numbers N {0,1, 2, · ..·} as the universe, where this sentence is true. (b) Build a model, using the natural numbers as the universe, where this sentence is false. (c) Is this sentence logically valid? Why or why not?

Let Z denote the set of integers. If m is a positive integer, we write Zm for the system of "integers modulo m." Some authors write Z/mZ for that system. For completeness, we include some definitions here. The system Zm can be represented as the set {0, 1,..., m - 1} with operations (addition) and (multiplication) defined as follows. If a, b are elements of {0, 1,..., m - 1}, define: ab the element c of {0, 1,...,m - 1} such that a +b-c is an integer multiple of m. a b = the element d of {0, 1,..., m - 1} such that ab -d is an integer multiple of m. For example, 30 4 = 2 in Z5, 303= 1 in Z4, and -1 = 12 in Z₁3. To simplify notations (at the expense of possible confusion), we abandon that new notation and write a + b and ab for the operations in Zm, rather than writing ab and a b. = Let Q denote the system of rational numbers. We write 4Z for the set of multiples of 4 in Z. Similarly for 4Z12. Consider the following number systems: Z, Q, 4Z, Z3, Z8, Z9, 4Z12, Z13. One system may be…