Suppose X and Y are nonempty sets and is an equivalence relation on X Y . Define

a relation on X by

x z y Y such that (x, y) (z, y)

(i) Is reflexive in general?

(ii) Is symmetric in general?

(iii) Is transitive in general?

In the above questions, "in general" means for all nonempty sets X, Y , and equivalence relations on X Y .

If your answer is 'yes', then you should give a proof. If your answer is 'no', then you should give an example of

X, Y , and where fails to have the specified property.

Question 2 (4 pts). Let A = Z \ {0} be the set of nonzero integers. Define the relation on A A by

(a, b) (c, d) abcd 0.

Show that is an equivalence relation.

Question 3 (3 pts). Prove that there are no integers x, y, z such that

9x⁶ + 13y⁵ + 4y² + 3z⁶ = 0 and 6x⁴ 2y⁵ + y² 3z⁸ = 1.

Hint: Prove the result by contradiction. Assume the equations have a solution and then do some computations

modulo 3 to arrive at a contradiction.

Question 4 (5 pts).

(i) Without using induction, prove that the following statement is true:

a Z (a² is divisible by 4 or a² + 3 is divisible by 4)

.

Hint: Use modular arithmetic.

(ii) Is the statement

( a Z, a² is divisible by 4 ) or ( a Z, a² + 3 is divisible by 4)

true? Justify your answer.

Question 5 (3 pts). Suppose p is a prime number. Prove that, for all x Z,

3x^p2 5x^p3+ 2x

is divisible by p.