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 ...

... 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 ...

... 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: ...

... (iii) (a) It is an equivalent relation since (1) Reflexivity: xRx, why? ... (2) Symmetry: xRy yRx, why? ... So yRx. (3) Transitivity: xRy and yRz implies that xRz. ...

... Combining transitivity and symmetry, we find that if ... So assume (b), namely, that ~ is reflexive and, for all a, b, c in G: if a ~ b and b ~ c, then c ~ a. ...