Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in the set of all positive integers:

This solution is comprised of a detailed explanation of the properties of the equivalence relation.
It contains step-by-step explanation of the following problem:

Decide which ones of the three properties of reflexivity, symmetry, and transitivity are true for each of the following relations in the set of all positive integers:

... x = 1. a. reflexive b. symmetric c. antisymmetric d. transitive. ... first coordinate 2 is not = 1. R is not reflexive. b. Symmetry means (x,y) ER implies (y,x) E R ...

... Equivalently, for all a, b and c in A: a ~ a. (Reflexivity) if a ~ b then b ~ a. (Symmetry) if a ~ b and b ~ c then a ~ c. (Transitivity) Now, we ...

... R is an equivalence relation, R is reflexive; Second, we ... R is an equivalence relation, R is transitive, therefore, if ... R c imply that a R c. By the symmetry of R ...

... Equivalently, for all a, b and c in A: ? a ~ a. (Reflexivity) ? if a ~ b then b ~ a. (Symmetry) ? if a ~ b and b ~ c then a ~ c. (Transitivity). ...

...Reflexivity: This is almost obvious, since b) Symmetry: This is ... (one example of relation reflexive but not ... c) Transitivity: this is to prove that According to ...

... lcm(a, b). b. Proof: Let T = R intersect S. Since both R and S are equivalence relations on X, then they satisfy the reflexivity, symmetry and transitivity. ...

... F is reflexive. Symmetry: Suppose that (a, b)F(c, d). Therefore ad = bc. But then cb = da, so (c, d)F(a, b). Therefore relation F is symmetric. Transitive: ...

Test Binary Relations. Test the binary relations on S for reflexivity, symmetry ,antisymmetry, and transitivity. A) S = Q X p Y <-> ABS(X) <= ABS(Y). ...