1. ## Equivalance Relations (congruences)

Hello everyone! How's it going ?

Could someone help me out with these equivalent relations questions? I'm kind of stuck.

For each of the following relations, determine whether it is reflexive, symmetric, and transitive and explain why in each case.

(1) Let G be the integers Z and let aRb if and only if |a − b| >= 5.
(2) Let G be the real numbers R and let xRy if and only if |x| + |y| = 1.
(3) Let G be the collection of all subsets of the set {0, 1, 2}, and let aRb if and only if each of a and b has the same number of members as the other.

Thank you!!

2. Originally Posted by clairepeterson
Hello everyone! How's it going ?

Could someone help me out with these equivalent relations questions? I'm kind of stuck.

For each of the following relations, determine whether it is reflexive, symmetric, and transitive and explain why in each case.

(1) Let G be the integers Z and let aRb if and only if |a − b| >= 5.
(2) Let G be the real numbers R and let xRy if and only if |x| + |y| = 1.
(3) Let G be the collection of all subsets of the set {0, 1, 2}, and let aRb if and only if each of a and b has the same number of members as the other.

Thank you!!

What've you done so far?

Tonio

3. Maybe to refresh your memory:
$\displaystyle \rho$ is reflexive means: $\displaystyle (\forall x)(x \in S \rightarrow (x,x) \in \rho)$
$\displaystyle \rho$ is symmetric means: $\displaystyle (\forall x)(\forall y)(x \in S \wedge y \in S \wedge (x, y) \in \rho \rightarrow (y, x) \in \rho)$
$\displaystyle \rho$ is transitive means: $\displaystyle (\forall x)(\forall y)(\forall z)(x \in S \wedge y \in S \wedge z \in S \wedge (x, y) \in \rho \wedge (y, z) \in rho \rightarrow (x, z) \in \rho)$

So, now you just have to use the formulars to check the relations:
1. reflexivity: $\displaystyle 5 \in G \rightarrow (5, 5) \in \rho$ or in other words for reflexivity $\displaystyle \Vert 5 - 5 \Vert \neq 5$.
symmetry: for example, $\displaystyle \Vert 5 - 10 \Vert = \Vert 10 - 5 \Vert$, but how do you show it for the whole set?
transitivity: $\displaystyle \Vert 5 - 10 \Vert = \Vert 10 - 15 \Vert \rightarrow \Vert 5 - 15 \Vert$ ??