1. Let X be a set and a binary relation on X .
a) Provide a mathematical de...nition for each of the following sentences:
- is re‡exive
- is transitive
- is symmetric
- is asymmetric
- is antisymmetric
- is a preorder
- is an equivalence relation
- X is chain
- X is a lattice
b) Are the following implications or their converse true? (Provide a coun- terexample whenever an implication is false.)
- is (not) symmetric =) is (not) asymmetric.
- is (not) symmetric =) is (not) antisymmetric.
- is (not) antisymmetric =) is (not) asymmetric.
2) Let X = f1; 2; 3; 12; 18; 36g be a set and a binary relation de...ned on
X as follows: for any x; y 2 X , x y if x is a multiple of y. Is X a lattice?
3) Let be a binary relation de...ned on a set X as follows: for any x; y 2 X ,
x y if x y 0 and x y is even.
a) Determine whether X is a lattice in each of the following situations:
a-i) X = f2; 4; 8g.
a-ii) X = f1; 4; 8; 9g.
b) Determine whether X is a chain in each of the following situations:
b-i) X = f2; 4; 8g.
b-ii) X = f1; 4; 8; 9g.
c) In each case, identify the maximal and the best elements.
4) An individual i is choosing a house from a set X of houses based on a set of n criteria, where his preference relation over X along each criterion is a complete preorder. From the viewpoint of i, criteria are ranked in order of importance (e.g., his ...rst criterion might be proximity to school, his second criterion proximity to the shopping center, and so on), and a house x is chosen over another house y if and only if x dominates y along the ...rst criterion in
which x and y di¤er. Denote by L
is a complete preorder.
the preference relation of i. Show that L
5) Let N = f1; 2; :::; ng be a set of soccer players and W P (N ) be the set of good or winning teams such that: for any teams S; T 2 P (N ), S 2 W and S T =) T 2 W (this means that adding more players to a good team always results in another good team). De...ne the desirability relation on N
as follows: let x; y 2 N be two players. x y if for any team S that contains neither x, nor y, S [ fyg 2 W =) S [ fxg 2 W (this means that x is a more desirable player than y if whenever y turns a bad team into a good team by joining it, x can do the same).
a) Is the indi¤erence component of an equivalence relation?
b) Is the irre‡exive component of transitive?
c) Is a preorder?
d) Provide an example where is not complete. e) Is N a lattice in general?
Assume that satis...es the following condition:
(*) For any good teams S; T 2 W and players x 2 S T and y 2 T S, S fxg [ fyg 2 W or T fyg [ fxg 2 W (this means that aone-to-one exchange of players between two good teams results in at least one of the resulting teams remaining good).
f ) Show that is complete.
g) Is condition (*) a necessary condition for the binary relation to be complete?
h) State a necessary and su¢cient condition for the binary relation to be a complete preorder.