Question

C. 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

Let's list the elements of R:

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

Find the domain  of the relation:

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

Find the range  of the relation:

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

Draw the digraph:

Find properties of the Relation:

$(5,5) \notin R$ so R is not reflexive

$(1,1) \in R$ so R is not irreflexive

If $x + y \le 6$ then $y+x \le 6$ so $\left( {x,y} \right) \in R \Leftrightarrow \left( {y,x} \right) \in R$ - R is symmetric relation.

$(5,1) \in R,\,(1,4) \in R$ , but $(5,4) \notin R$ so R is not transitive relation.