In this question all sets are finite. Let N {0, . . . , 9}^7 be a set of perm nu

Question

In this question all sets are finite. Let N {0, . . . , 9}^7 be a set of perm numbers, S a set of student names, and C a set of courses. Let F N × S be the function that maps each perm number to the name of the corresponding student such that (n, s) F if and only if the name of student with perm number n is s, and let G C×N be the relation that describes which students are attending which courses, such that (c, n) G if and only if the student with perm number n takes course c. Using N, S, C, F, and G, give a formal definition of the set Z S^2 of pairs of names of different students that are jointly enrolled in at least two courses. In other words, (s, t) Z if and only if s and t are both enrolled in two different courses. Your formal definition should be given in a rigorous mathematical manner that does not use English. Note also that different students (with different perm numbers) can have the same name. Briefly explain, in English, your construction.

Explanation / Answer

N {0, . . . , 9}^7 be a set of perm numbers, S a set of student names, and C a set of courses

F N × S be the function that maps each perm number to the name of the corresponding student such that (n, s) F if and only if the name of student with perm number n is s.

G C×N be the relation that describes which students are attending which courses, such that (c, n) G if and only if the student with perm number n takes course c.

set Z S^2 of pairs of names of different students that are jointly enrolled in at least two courses.

For a student in Z must have a perm number and two courses atleast.
so the equation (s,t) Z will true only when
there is a s such that (n1, s) F and (c1,n1) G and (c2,n1) G. where n1 is the perm number of student s and c1 and c2 are courses taken by the student.

Similalry for t we must have (n2, t) F and (c1,n2) G and (c2,n2) G. where n2 is the perm number of student t and c1 and c2 are courses taken by the student.

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