Each of the following defines a relation on N:
x + y = 10, x, y ∈N

Determine which of the above relations are reflexive, symmetric and transitive.

Question

Each of the following defines a relation on N: x + y = 10, x, y ∈ N Determine which of the above relations are reflexive, symmetric and transitive.

Philoid · Accepted Answer

Let R = {(x,y): x + y =10, ∀ x,y ∈ N } be a relation defined on N.

R = {(1,9),(2,8),(3,7),(4,6),(5,5),(6,4),(7,3),(8,2),(9,1)}

Now,

R is not reflexive ∵ (2,2) ∉ R.

R is symmetric ∵ (3,7) ∈ R ⇒ (7,3) ∈ R

R is not transitive ∵ (1,9) ∈ R and (9,1) ∈ R but (1,1) ∉ R.

Thus, R is symmetric but not reflexive not transitive.

Each of the following defines a relation on N:x + y = 10, x, y ∈NDetermine which of the above relations are reflexive, symmetric and transitive.