Question 1
Question on equivqlence relation

Let S={1,2,3,4,5}. Define R by
 
    xRy <==> x-y=2k for some integer k.

Then R can be given by listing up the pairs that satisfy R.

R={(1,1), (2,2), (3,3), (4,4), (5,5), (1,3), (3,1), (3,5), (5,3)
   (1,5), (5,1), (2,4), (4,2)}

Equivalence classes are given by {1,3,5}, {2,4}

Following this example, solve the same question for

    S={1,2,3,4,5,6,7}
    xRy <==> x-y=3k for some integer k

Question 2
We consider functions from set S to set S in the following.
Let S={0,1,2,3,4} and define f(x)=(x+2) mod 5. Then f is given by
the set of pairs {(0,2), (1,3), (2,4), (3,0), (4,1)}. This is
a bijection. 
  Now let S={0,1,2,3,4}, and define f(x)=(x^2) mod 3. Following the above
exapmle, obtain f in the form of set of pairs. Is this function a bijection?
If not, what type of function is it?

Question 3. We consider permutations on (1,2,...,n).
The set of permutations in which 1 is not at the first position with
n=3 is given by {(2,1,3), (2,3,1), (3,1,2), (3,2,1)}. In general
there are (n-1)(n-1)! such permutations. Following this example,
list up all permutations such that 2 is not at the second position
with n=4.
 
Question 4. Logic
Are following logic formulae true or false? Give the reason.

(1) (for some x)(for all y) x<2y
(2) (for all x)(for some y) x<2y
(3) (for some x)(for some y) x<2y

For other questions, study tutorial problems carefully.