Question

A relation is defined on + × + by (m,)(p,) ⟺ m=p .

(a). Prove that is an equivalence relation.

(b). Use the set builder notation to describe the equivalence class [(1,2)].

Answer 1

The relation defined on the set is an equivalence relation because it satisfies reflexivity, symmetry, and transitivity. The equivalence class [(1,2)] consists of all pairs (p,q) where p=1.

Step-by-step explanation:

To determine whether a relation ⟡ defined on + × + by (m,n) ⟡ (p,q) ⟡ m=p is an equivalence relation, we must show that it satisfies three properties: reflexivity, symmetry, and transitivity.

• Reflexivity: For any (m,n) ∈ + × +, obviously m=m. Therefore, (m,n) ⟡ (m,n), satisfying reflexivity.
• Symmetry: If (m,n) ⟡ (p,q), then m=p. If we reverse the order, meaning if we have (p,q) ⟡ (m,n), then p=m. Since m=p, we also have p=m, satisfying symmetry.
• Transitivity: If (m,n) ⟡ (p,q) and (p,q) ⟡ (r,s), then m=p and p=r, thus m=r. Hence, (m,n) ⟡ (r,s), satisfying transitivity.

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

To describe the equivalence class [(1,2)] using set builder notation, we consider all pairs (p,q) that relate to (1,2). Since the relation is defined by the first component being equal, the equivalence class is all pairs where the first component is 1. Thus, [(1,2)] = p=1.