9. (9 points) Define a relation R on Z as follows: For all integers m and n, m Rn ↔m>n. a) Find two pairs (m, n) which satisfy R. b) Show that R is not reflexive. c) Show that R is transitive. Do NOT just list numerical examples.

please answer fully on paper and make sure its correct please i keep getting different answers

Let R be a relation on the set of all integers such that aRb if and only if a + b is even. To show that R is symmetric, we must prove that For any integers a and b, if a + b is odd, then b + a is even O For any integers a and b, if a + b is even, then b + a is even O For any integers a and b, if a + b is odd, then b + a is odd O For any integers a and b, if a + b is even, then b + a is odd To show that R is not symmetric, we must find a counterexample satisfies which of the following statement? O For some integers a and b, a + b is even and b + a is odd O For some integers a and b, a + b is even and b + a is even O For some integers a and b, a + b is odd and b + a is odd O For some integers a and b, a + b is odd or b + a is even Note: In order to get credit for this problem all answers must be correct.

Define a relation S on R, the set of all real numbers as follows: For every x, y ∈ R, x S y ⇐⇒ x 2 ≤ y 2 . Determine whether S is a partial order relation. Prove or give a counterexample.