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2. (5pt) Consider the following binary relations. In each case prove the relation in question is...
2. (5pt) Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are. (a) Si is a binary relation on R2 x R2 defined by z+ly-+ 1 r,y). (,y) e S Recall that R =R x R. (b) Sa is a binary relation on R defined by 1-ye2 r,y) e S
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define...
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define mn if and only if the sum of the digits (b) On the set (1.2,3,4,11, 12, 13,14,21,22,23,24), define mn if and only diagram for the induced partial order on the equivalence classes of m is less than or equal to the sum of the digits of n. if...
Can you #2 and #3? 6. LESSON 6 (1) Let A be the set of people alive on earth. For each relation defined below, determin...
Can you #2 and #3?
6. LESSON 6 (1) Let A be the set of people alive on earth. For each relation defined below, determine if it is an equivalence relation on A. If it is, describe the equivalence classes. If it is not determine which properties of an equivalence relation fail. (a) a Hb a and b are the same age in (in years). (b) a Gb a and b have grandparent in common. 2) Consider the relation S(x,y):x...
Problem 3. Which of the following relations are equivalence relations on the given set S (1)...
Problem 3. Which of the following relations are equivalence relations on the given set S (1) S-R and ab5 2a +3b. (2) S-Z and abab 0 (4) S-N and a~b ab is a square. (5) S = R × R and (a, b) ~ (x, y)-a2 + b-z? + y2.
10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2...
10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2 - y2 = x - y (a) Show that R is reflexive. (b) Show that R is symmetric. (c) Show that R is transitive. (d) You have thus verified that R is an equivalence relation. What is the equivalence class of 3? (e) More generally, what is the equivalence class of an element x? Use the listing method. (f) Instead of proving the three...
2. A binary string is a finite sequence u-діаг . . . an, where each ai is either 0 or 1. In this case n is the length o...
2. A binary string is a finite sequence u-діаг . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings ai, aia2,... ,ai... an-1,ai... an are all prefixes of v. On the set X of all binary strings consider the relations Ri and R2 defined as follows: Ri-(w, v) w and v have the same length ) R2 = {(u, v) I w is a prefix...
Determine whether each of the relations defined below on the set of positive integers is a...
Determine whether each of the relations defined below on the set of positive integers is a partial order? (a) (x,y) ϵ R if x ≥ y (b) (x,y) ϵ R if 3 divides x+y
1. (14 points) (a) For each of the following relations R on the given domains A,...
1. (14 points) (a) For each of the following relations R on the given domains A, categorize them as not an equivalence relation, an equivalence relation with finitely many distinct equivalence classes, or an equivalence relation with infinitely many distinct equivalence classes. Justify each decision with a brief proof. (i) A = {1, 2, 3} , R = {(1, 1),(2, 2),(3, 3)} (ii) A = R, R = {(x, y) | x 2 = y 2} (iii) A = Z,...
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each...
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each of these statements. (a) RUS is reflexive. (b) Rn S is reflexive. (c) R\S is reflexive. (2) Define the equivalence relation on the set Z where a ~b if and only if a? = 62. (a) List the element(s) of 7. (b) List the element(s) of -1. (c) Describe the set of all equivalence classes.
2. A binary string is a finite sequence v = a1a2 . . . an, where each ai is either 0 or 1. In this case n is the length...
2. A binary string is a finite sequence v = a1a2 . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings a1, a1a2, . . . , a1 . . . an−1, a1 . . . an are all prefixes of v. On the set X of all binary strings consider the relations R1 and R2 defined as follows: R1 = {(w, v) | w...
2. (5pt) Consider the following binary relations. In each case prove the relation in question is an equivalence relation and describe, in geometric terms, what the equivalence classes are. (a) Si is a binary relation on R2 x R2 defined by z+ly-+ 1 r,y). (,y) e S Recall that R =R x R. (b) Sa is a binary relation on R defined by 1-ye2 r,y) e S
10. Verify that the relations given below are quasiorders. List the elements of each equivalence class of the induced equivalence relation, and draw the Hasse (a) On the set (1,2,..., 303, define mn if and only if the sum of the digits (b) On the set (1.2,3,4,11, 12, 13,14,21,22,23,24), define mn if and only diagram for the induced partial order on the equivalence classes of m is less than or equal to the sum of the digits of n. if...
Can you #2 and #3?
6. LESSON 6 (1) Let A be the set of people alive on earth. For each relation defined below, determine if it is an equivalence relation on A. If it is, describe the equivalence classes. If it is not determine which properties of an equivalence relation fail. (a) a Hb a and b are the same age in (in years). (b) a Gb a and b have grandparent in common. 2) Consider the relation S(x,y):x...
Problem 3. Which of the following relations are equivalence relations on the given set S (1) S-R and ab5 2a +3b. (2) S-Z and abab 0 (4) S-N and a~b ab is a square. (5) S = R × R and (a, b) ~ (x, y)-a2 + b-z? + y2.
10. [12 Points) Properties of relations Consider the relation R defined on R by «Ry x2 - y2 = x - y (a) Show that R is reflexive. (b) Show that R is symmetric. (c) Show that R is transitive. (d) You have thus verified that R is an equivalence relation. What is the equivalence class of 3? (e) More generally, what is the equivalence class of an element x? Use the listing method. (f) Instead of proving the three...
2. A binary string is a finite sequence u-діаг . . . an, where each ai is either 0 or 1. In this case n is the length of the string v. The strings ai, aia2,... ,ai... an-1,ai... an are all prefixes of v. On the set X of all binary strings consider the relations Ri and R2 defined as follows: Ri-(w, v) w and v have the same length ) R2 = {(u, v) I w is a prefix...
(1) Suppose R and S are reflexive relations on a set A. Prove or disprove each of these statements. (a) RUS is reflexive. (b) Rn S is reflexive. (c) R\S is reflexive. (2) Define the equivalence relation on the set Z where a ~b if and only if a? = 62. (a) List the element(s) of 7. (b) List the element(s) of -1. (c) Describe the set of all equivalence classes.
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