Let S be a set, and R an antisymmetric relation on S. Prove that R^c is trichotomous.
I don't thing the given statement is true because you just consider the antisymmetric relation divisibility on natural numbers. Then compliment of this relation is
a is related to b iff a does not divde b . This is not a trichotomous relation because
Hence is not trichotomous
Let S be a set, and R an antisymmetric relation on S. Prove that R^c is...
Let R be a relation on a set A. Prove that R is antisymmetric if and only if R ∩ R ^(−1) ⊆ {(a, a) : a ∈ A}.
[Partial Orders - Six Easy Pieces] A binary relation is R is said to be antisymmetric if (x,y) ER & (y,x) ER = x=y. For example, the relations on the set of numbers is antisymmetric. Next, R is a partial order if it is reflexive, antisymmetric and transitive. Here are several problems about partial orders. (a) Let Ss{a,b} be a set of strings. Let w denote the length of the string w, i.e. the number of occurrences of letters (a...
Problem 2: Let R,SCAx A be antisymmetric relations. Prove that the union RUS is antisymmetric if and only if ROSCIA Problem 2: Let R,SCAx A be antisymmetric relations. Prove that the union RUS is antisymmetric if and only if ROSCIA
probelms 9.1 9 Modular arithmetic Definition 9.1 Let S be a set. A relation R = R(,y) on S is a statement about pairs (x,y) of elements of S. For r,y ES, I is related to y notation: Ry) if R(x,y) is true. A relation Ris: Reflexive if for any I ES, R. Symmetric if for any ry ES, Ry implies y Rr. Transitive if for any r.y.ES, Ry and yRimply R. An equivalence relation is a reflexive, symmetric and...
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.
4. Let S be the set of continuous function f: [0;1) ! R. Let R be the relation defined on S by (f; g) 2 Rif(x) is O(g(x)). (a) Is R reflexive? (b) Is R antisymmetric? (c) is R symmetric? (d) is R transitive? Explain your answer in details. Use the definition of big-O to justify your answer if you think R has a certain property or give a counter example if you think R does not have a certain...
(17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f is 0(g)" on F is: (a) (4pt) Write down the definition for "f is O(g)". (b) (4pt) Prove that the relation is reflexive (c) (6pt) Prove that the relation is not symmetric. (d) (6pt) Prove that the relation is transitive. (17) (20pt) Let F be the set of functions f : R+ → R. Prove that the binary relation "f...
Consider the empty set as a relation, R, on any non-empty set S. Prove or disprove: R is transitive.
1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the set S defined as follows: Va,bE S, arb if and only if every prime number that divides a is a factor of b and a S b. The relation T is a partial order relation (you do not need to prove this). Draw the Hasse diagram for T 1. (2 marks) Let S 2,3,4,5,6,7,8,9, 10, 11, 12). Let r be the relation on the...
Let R be a relation from A to B and S be a relation from B to C . Prove a. (s')'=s b. (S-R)'ER'.51