Each of the following defines a relation on N:
x + 4y = 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 + 4y = 10 x, y ∈ N. Determine which of the above relations are reflexive, symmetric and transitive.

Philoid · Accepted Answer

Let R = {(x,y): x + 4y =10, ∀ x,y ∈ N } be a relation defined on N.

R = {(2,2),(6,1)}

Now,

R is not reflexive ∵ (1,1) ∉ R.

R is not symmetric ∵ (6,1) ∈ R but (1,5) ∉ R

R is not transitive

∵ (x,y) ∈ R ⇒ x+4y=10 and (y,z) ∈ R ⇒ y+4z=10

⇒ x-16z = -30

⇒ (x,z) ∉ R

Thus, R is neither symmetric nor reflexive nor transitive.

Each of the following defines a relation on N:x + 4y = 10 x, y ∈N.Determine which of the above relations are reflexive, symmetric and transitive.