Question

RELATION.

Given the following set:

1. X = {1, 2, 3, 4, 5} defined by the rule (x, y) ∈ R if x + y ≤ 6 

a. List the elements of R

b. Find the domain of R

c. Find the range of R

d. Draw the digraph

e. Properties of the Relation 

Answer 1

Given the $X = \{1, 2, 3, 4, 5\}$ define the relation $R$ by the rule $(x, y) \in R$ if $x + y \le 6$  

a. Let us find the list the elements of $R:$

$R=\{(1,1),(1,2),(2,1),(1,3),(3,1),(2,2),(1,4),(4,1),(2,3),(3,2),(1,5), \newline (5,1),(2,4),(4,2),(3,3)\}$

b. Let us find the domain of $R:$

$dom(R)=\{x\in X\ |\ (x,y)\in R\}=\{1,2,3,4,5\}$

c. Let us find the range of $R:$

$range(R)=\{y\in X\ |\ (x,y)\in R\}=\{1,2,3,4,5\}$

d. Let us draw the digraph of $R:$

e. Let us study some properties of the relation $R:$

Since $(4,4)\notin R$, the relation $R$ is not reflexive.

If $(x, y) \in R$, then $x + y \le 6$, and hence $y + x \le 6$. It follows that $(y, x) \in R$, and the relation is symmetric.

Taking into account that $(5,1)\in R$ and $(1,5)\in R$, but $(5,5)\notin R$, we conclude that $R$ is not a transitive relation.

It follows that $R$ is not an equivalence relation.