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Let X = R × R. We define the preference relation R on X, where (a,...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0,...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0, 1) and oo > n > 2, with x E C = Ri. Answer thefollow (x1+x2)" ing: a. Show the preference relation that this utility function induces "upper b. Show the preference relation these preferences represent are strictly C. Give another utility function that generates exactly the same behavior as level sets that are convexif U(x) is Convex for any xeX monotonic. this one....
4.1 6b Let A be the set {a,b,c}, and define a relation on A as R...
4.1 6b Let A be the set {a,b,c}, and define a relation on A as R = {(x,y) E AXA : 2x + y is prime}. Prove that R is a function with domain A.
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on...
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
2. Assume a consumer has as preference relation represented by u(x1, x2) = axi + bx2...
2. Assume a consumer has as preference relation represented by u(x1, x2) = axi + bx2 with x E C = R4, and a, b > 0. Answer the following: a. Show the preference relation this consumer is convex and strictly monotonic show preferences are not strictly convex for this consumer. b. Graph the indifference curves for this consumer. Now, solve for an explicit expression for the indiffence curve (i.e., x (x1; ū) i constructed in class for an indifference...
Define a relation R from R to R as follows: For all (x, y) E R...
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f (y). a. Prove that R is an equivalence relation on A. b. Let Ex = fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x 2 Ag...
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only if either x-a or y- b. Prove or disproof, isan equivalence relation? If so, write down all the equivalence classes.
Let z denote a complete, reflexive and transitive weak preference relation over a set X, and...
Let z denote a complete, reflexive and transitive weak preference relation over a set X, and let > denote the strict preference relations derived from 2. Select one: O a. the strict preference relation is neither transitive nor complete. O b. the strict preference relation is both transitive and complete. c. the strict preference relation is transitive but not necessarily complete. O d. the strict preference relation is complete but not necessarily transitive.
Let A = ( a, b, c, d ) and let ( A, R ) be a posset where R is a Relation on A defined by: ...
Let A = ( a, b, c, d ) and let ( A, R ) be a posset where R is a Relation on A defined by: R is reflexive c ≤ d a ≤ c a ≤ b a ≤ d b ≤ d Find H(A) Is (A, R) a lattice? If you answer no, give a counterexample. If you answer yes, give a brief justification as to why (no formal proof needed). Is (A,R) a Boolean algebra? Give...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for ...
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
1. Assume a consumer has as preference relation represented by u(c1, 2) for g E (0, 1) and oo > n > 2, with x E C = Ri. Answer thefollow (x1+x2)" ing: a. Show the preference relation that this utility function induces "upper b. Show the preference relation these preferences represent are strictly C. Give another utility function that generates exactly the same behavior as level sets that are convexif U(x) is Convex for any xeX monotonic. this one....
4.1 6b Let A be the set {a,b,c}, and define a relation on A as R = {(x,y) E AXA : 2x + y is prime}. Prove that R is a function with domain A.
[12] 5. Let A = {1, 2, 3, 4, ..., 271}. Define the relation R on A x A by: for any (a,b), (c,d) E AXA, (a,b) R (c,d) if and only if a +b=c+d. (a) Prove that R is an equivalence relation on AX A. (b) List all the elements of [(3,3)], the equivalence class of (3, 3). (c) How many equivalence classes does R have? Explain. (d) Is there an equivalence class that has exactly 271 elements? Explain.
2. Assume a consumer has as preference relation represented by u(x1, x2) = axi + bx2 with x E C = R4, and a, b > 0. Answer the following: a. Show the preference relation this consumer is convex and strictly monotonic show preferences are not strictly convex for this consumer. b. Graph the indifference curves for this consumer. Now, solve for an explicit expression for the indiffence curve (i.e., x (x1; ū) i constructed in class for an indifference...
Define a relation R from R to R as follows: For all (x, y) E R x R, (x, y) E R if, and only if, x= y2 + 1. (a) Is (2, 5) E R? Is (5, 2) e R? Is (-3) R 10? Is 10 R (-3)? (b) Draw the graph of R in the Cartesian plane. (C) Is R a function from R to R? Explain.
2. Let f : A ! B. DeÖne a relation R on A by xRy i§ f (x) = f
(y). a. Prove that R is an equivalence relation on A. b. Let Ex =
fy 2 A : xRyg be the equivalence class of x 2 A. DeÖne E = fEx : x
2 Ag to be the collection of all equivalence classes. Prove that
the function g : A ! E deÖned by g (x) = Ex is...
Problem 5. Define a relation ~on R x R as (x, y) ~(a,b) if and only if either x-a or y- b. Prove or disproof, isan equivalence relation? If so, write down all the equivalence classes.
Let z denote a complete, reflexive and transitive weak preference relation over a set X, and let > denote the strict preference relations derived from 2. Select one: O a. the strict preference relation is neither transitive nor complete. O b. the strict preference relation is both transitive and complete. c. the strict preference relation is transitive but not necessarily complete. O d. the strict preference relation is complete but not necessarily transitive.
2. Let S 11,2,3,4,5, 6, 7,8,91 and let T 12,4,6,8. Let R be the relation on P (S) detined by for all X, Y E P (s), (X, Y) E R if and only if IX-T] = IY-T]. (a) Prove that R is an equivalence relation. (b) How many equivalence classes are there? Explain. (c) How mauy elements of [ø], the equivalence class of ø, are there? Explain (d) How many elements of [f1,2,3, 4)], the equivalence class of (1,2,3,...
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