Question

. Check whether the relation R in the set Z of integers defined as
R = {(a, b): a + b is "divisible by 2"} is reflexive, symmetric or transitive.​

Answer 1

Question :-

Check whether the relation R in the set Z of integers defined as R = {(a,b): a + b is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].

Reflexive :

Since, a + a = 2a which is even∴ (a,a) ∈ R ∀a ∈ Z

Hence R is reflexive

Symmetric :

If (a,b) ∈ R, then a + b = 2λ ⇒ b + a = 2λ

⇒ (b,a) ∈ R,

Hence R is symmetric

Transitive :

If (a,b) ∈ R and (b,c,) ∈ R

Then a + b = 2λ ---(1) and b + c = 2μ --- (2)

Adding (1) and (2) we get

⇒a + 2b + c = 2(λ + μ)

⇒ a + c = 2(λ + μ − b)

⇒a + c = 2k, where λ + μ − b = k ⇒ (a,c) ∈ Hence R is transitive.

[0] = {...-4, -2, 0, 2, 4...}

Answer 2

Answer:

Question :-

Check whether the relation R in the set Z of integers defined as R = {(a,b): a + b is "divisible by 2"} is reflexive, symmetric or transitive. Write the equivalence class containing 0 i.e. [0].

Reflexive :

Since, a + a = 2a which is even∴ (a,a) ∈ R ∀a ∈ Z

Hence R is reflexive

Symmetric :

If (a,b) ∈ R, then a + b = 2λ ⇒ b + a = 2λ

⇒ (b,a) ∈ R,

Hence R is symmetric

Transitive :

If (a,b) ∈ R and (b,c,) ∈ R

Then a + b = 2λ ---(1) and b + c = 2μ --- (2)

Adding (1) and (2) we get

⇒a + 2b + c = 2(λ + μ)

⇒ a + c = 2(λ + μ − b)

⇒a + c = 2k, where λ + μ − b = k ⇒ (a,c) ∈ Hence R is transitive.

[0] = {...-4, -2, 0, 2, 4...}

Step-by-step explanation:

Hope this answer will help you.