Problem 2: Let R,SCAx A be antisymmetric relations. Prove that the union RUS is antisymmetric if ...
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 S be a set, and R an antisymmetric relation on S. Prove that R^c is trichotomous.
[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 the following.
Let R and S be relations on a set X, and let A C X Prove the following except when asked to give a coun terexample If R and S are both transitive, then RnS is tran- sitive
(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.
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a) R-(x, y):x +2y 3), defined on the set A 10, 1,2,3) (b) R-I(x, y): xy 4), defined on the set A (0,1,2,3,4 (c) R-(x, y): xy 4), defined on the set A-0,,2,3)
Question 2 For each of the following relations R, determine (and explain) whether R is: (1) reflexive (2) symmetric (3) antisymmetric (4) transitive (a)...
1. (9 points; 3 points each) symmetric, antisymmetric, and transitive. (2) R ((a, b) | la-bl 23) the following relations on the set of integers and indicate whether each is refexive,
1. (9 points; 3 points each) symmetric, antisymmetric, and transitive. (2) R ((a, b) | la-bl 23) the following relations on the set of integers and indicate whether each is refexive,
Problem set 9 (10 marks). Let K be a KC UFENI The aim of this exercise is to prove that there is n finite union of the open intervals) compact set of R and (I,)rEN be open intervals such that N such that K C I U..U (i.e. K is actually contained in a n E N, select a, K such that 1. Assume that the result does not hold, and explain why we can then, for any n UUIn...
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,...
Let U cR. Prove that U is the union of countably many disjoint open intervals. Aryue first that U is the union of disjoint intervals by "joining together" neighborhoods that overlap (make this precise!). Then argue that is Q is dense in R, there are at most countably many such intervals