Let T be a relation on R x R such that (a,b),(c,d) ∈ T if and only if a + b ≤ c + d.

a. Is T reflexive? If so, prove it. If not, prove that it is not.

b. Is T symmetric? If so, prove it. If not, prove that it is not.

c. Is T transitive? If so, prove it. If not, prove that it is not.

d. Is T an equivalence relation? If so, explain why it is. If not, explain why not.

e. Identify four numbers of [(-1, 2)]

5. Define a relation R on the set of real numbers by (x, y) ER if and only if |1 – y <3. Is this relation reflexive? Is it symmetric? Is it transitive? Is this an equivalence relation? Justify your arguments.

5. Define a relation on Q-{0} as follows: In -= 2for some keZ. Prove that this is an equivalence relation.

Does the relation "is parallel to" have a

a. reflexive property? (consider a line )

b. symmetric property? (consider lines and in a plane)

c. transitive property? (consider coplanar lines , , and )