Let A = {1, 2, 3, 4, 5, 6) and let R = {(a, b) : a, b ∈ A and b = a + 1}. Show that R is (i) not reflexive, (ii) not symmetric and (iii) not transitive.

Question

Philoid · Accepted Answer

Given that,

A = {1, 2, 3, 4, 5, 6) and R = {(a, b) : a, b ∈ A and b = a + 1}.

∴ R = {(1,2),(2,3),(3,4),(4,5),(5,6)}

Now,

R is Reflexive if (a,a) ∈ R ∀ a ∈ A

Since, (1,1),(2,2),(3,3),(4,4),(5,5),(6,6) ∉ R

Thus, R is not reflexive .

R is Symmetric if (a,b) ∈ R ⇒ (b,a) ∈ R ∀ a,b ∈ A

We observe that (1,2) ∈ R but (2,1) ∉ R .

Thus, R is not symmetric .

R is Transitive if (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R ∀ a,b,c ∈ A

We observe that (1,2) ∈ R and (2,3) ∈ R but (1,3) ∉ R

Thus, R is not transitive.

Let A = {1, 2, 3, 4, 5, 6) and let R = {(a, b) : a, b ∈ A and b = a + 1}.Show that R is (i) not reflexive, (ii) not symmetric and (iii) not transitive.

Let A = {1, 2, 3, 4, 5, 6) and let R = {(a, b) : a, b ∈ A and b = a + 1}.
Show that R is (i) not reflexive, (ii) not symmetric and (iii) not transitive.