Question

Proof that over Q, the relation: a ∼ b if a - b ∈ ℤ is an equivalence relation over Q. This relation is called congruence relation modulo ℤ. What is the question related to this?

Answer 1

To prove that the relation a ⊼ b (a - b ∈ ℤ) is an equivalence relation over Q, we need to show that it satisfies the properties of reflexivity, symmetry, and transitivity.

Step-by-step explanation:

In order to prove that the relation a ⊼ b (a - b ∈ ℤ) is an equivalence relation over Q, we need to show that it satisfies three properties: reflexive, symmetric, and transitive.

Reflexive Property:

For any number a ∊ Q, a - a = 0 which is an integer. Therefore, a ⊼ a and the relation is reflexive.

Symmetric Property:

If a ⊼ b, then a - b is an integer. And if a - b is an integer, then -(a - b) = b - a is also an integer. Hence, if a ⊼ b, then b ⊼ a and the relation is symmetric.

Transitive Property:

If a ⊼ b and b ⊼ c, then (a - b) + (b - c) = a - c which is an integer. Therefore, if a ⊼ b and b ⊼ c, then a ⊼ c and the relation is transitive.

Since the relation satisfies all three properties, it is an equivalence relation over Q.