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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.
Determine the stable and unstable manifolds for the rest point of the system.
A group of ducks and cows are in a field. They have a total of 65 heads and 226 legs. How many ducks are in the field?
Probability : Combinations, Mr james just won 6 tickets for each of 2 consecutive Giants home football games. For the first game, Mrs James will not be able to attend so he has 5 extra tickets.
consider the curve x^2+sin(xy)+3y^2=c, where c is constant. the point (1,1) lies on the curve. use tangent line approximation to approximate the y coordinate when x=1.01
Use shells to find the volume of the solid formed by revolving the region bounded by the x-axis and y=4x-x^2 about the y-axis. Sketch the area. Include a typical rectangle.
A dance team knows 17 routines of which 9 are tap, 5 are ballet, and 3 are modern. The program can consist of any five routines. In how many ways can the manager choose which routines to present if there are no restrictions?
Suppose that we toss balls into b bins until some bin contains two balls. Each toss is independent, and each ball is equally likely to end up in any bin. What is the expected number of ball tosses?
Test hypothesis for variance.
write your answer in words, labeling units as necessary. Hank bicycles 5 km/hour slower than Kelly. In the time that it takes Hank to bicycle 42 km, Kelly can bicycle 57 km. How fast does each bicyclist travel?
a ski lift that runs to the top of a hill has a rise of 1/9. The horizonal distance from the bottom of the lift to the center of the mountain is 7000 feet. How hight is the hill?
Suppose you have a lemonade stand, and when you charge $2.00 per cup of lemonade and you sell 100 cups. But when you raise your price to $4.00 you only sell 50 cups
Find an equation for the planet's orbit. (Place the origin at the center of the orbit with the sun on the x-axis.
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