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}.
Let R be a relation on a set A. Prove that R is antisymmetric if and...
Let S be a set, and R an antisymmetric relation on S. Prove that R^c is trichotomous.
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
[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...
Prove that if R is an equivalence relation on a set A, then R ^-1 is an equivalence relation on A.
(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...
Discrete Mathematics. Let A = {2,3,4,6,8,9,12,18}, and define a relation R on A as ∀x,y ∈ A,xRy ↔ x|y. (a) Is R antisymmetric? Prove, or give a counterexample. (b) Draw the Hasse diagram for R. (c) Find the greatest, least, maximal, and minimal elements of R (if they exist). (d) Find a topological sorting for R that is different from the ≤ relation.
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 the relation on the set of ordered pairs of positive integers such that ((a, b), (c, d)) Element R if and only if ad = bc. Show that R is an equivalence relation What is the equivalence class of of (1, 2), i.e. [(1, 2)]?
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.
Prove that the following relation R is an equivalence relation on the set of ordered pairs of real numbers. Describe the equivalence classes of R. (x, y)R(w, z) y-x2 = z-w2