Relations assignment

1. Prove or disprove that < ("less than") is a partial order on .

2. Determine whether {(a,a)(a,b),(b,b),(c,c),(c,d),(c,e),(d,d),(d,e), (e,e)} is a partial order on {a,b,c,d,e}.

3. Prove or disprove that "is a subset of" is a partial order on the power set of a set A.

4.For a poset({2,4,6,9,12,18,27,36,48,60,72}, |), answer the following.

A. Give all maximal elements.

B. Give all minimal elements.

C. Is there the greatest element? If so, what is it?

D.Is there the least element? If so, what is it?

E. Give all upper bounds of {2, 9}.

F. Is there the least upper bound of {2, 9}? If so, what is it?

G. Give all lower bounds of {60, 72}.

H. Is there the greatest lower bound of {60, 72}? If so, what is it?

5. Let R(S) denote the set of all binary relations on a set S. A binary relation < on R(S) is defined as R 1 < R2 iff R 1 R2.Prove or disprove that (R(S), < ) is a poset.

6. Let be an alphabet. Assume that is totally ordered with a total order on .Let < denote a lexicographic order on.Prove or disprove that (, < ) is a poset.

7. Let | be a binary relation on + such that a | b iff b = ka for some integer k. Prove or disprove that (+, | ) is a poset.

8. Let | be a binary relation on such that a | b iff a 0 and b = ka for some integer k. Prove or disprove that (, | ) is a poset.

11. Let R be a binary relation on + such that a R b iff a b and b = ka for some integer k. Prove or disprove that R has each of the following 3 properties: Reflexivity, antisymmetry and transitivity.

12. Let R be abinary relation on such that (x1, y 1) R (x2 ,y2) iff ( x1 < x2) ( x 1= x 2 y 1 y 2). Prove or disprove that (2,R) is a poset.

13. Let R be a binary relation on + such that x R y iff x y ( x ylcm(x,y) = xy ), where "lcm" stands for the least common multiple. Prove or disprove that (+, R) is a poset.