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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.
What is the p-value associated with the test of hypothesis you conducted?
Coffee is draining - Application of calculus
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Find the solution of the initial value problem
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solve equation radio the instructor has learned that for the 30 students in her afternoon class, the radio of student who do not own a computer to those who do is 2 to 1. how many students in the class own their own computer?
If the test is positive, what is the probability that the person actually has hepatitis? (Round your answer to five decimal places.)
The pipeline will be built in a straight line from the rig to a selected point on the shoreline, then in a straight line to the storage tank. What point on the shoreline should be selected to minimize the total cost of the pipeline?
Pasta comes with tomato sauce and can be ordered with some , all, or none of these ingredients in the sauce: {onions, garlic, carrots, broccoli,shrimp, mushrooms, zucchini, green pepper}.
A flagpole sitting on a level field is 30 feet tall. the flagpole casts a sahdow on the ground that is 18 feet long. A stick placed upright in the ground has a height of 5 fett. what is the length in feet of the stick's shadow.
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