Answer:
Z = {even integers} ⋃ {odd integers}
Each subset in the partition contains elements that are related to each other under the ∼ relation. Elements within the same subset have an even sum when added, while elements from different subsets have an odd sum when added.
Explanation:
(a) To prove that ∼ defines an equivalence relation on Z, we need to show that it satisfies three properties: reflexivity, symmetry, and transitivity.
1. Reflexivity: For any integer a ∈ Z, we need to show that a ∼ a. In other words, we need to prove that a + a is even. Since a + a = 2a, where a is an integer, we can see that a + a is even by definition. Therefore, the relation is reflexive.
2. Symmetry: For any integers a and b in Z, if a ∼ b, then we need to show that b ∼ a. If a + b is even, we have to prove that b + a is also even. Since addition is commutative, b + a = a + b, and if a + b is even, then b + a is also even. Therefore, the relation is symmetric.
3. Transitivity: For any integers a, b, and c in Z, if a ∼ b and b ∼ c, then we need to show that a ∼ c. If a + b is even and b + c is even, we have to prove that a + c is even. We know that a + b is even, so there exists an integer k1 such that a + b = 2k1. Similarly, b + c is even, so there exists an integer k2 such that b + c = 2k2. Adding these two equations, we have a + b + b + c = 2k1 + 2k2, which simplifies to a + c + 2b = 2(k1 + k2). Since k1 + k2 is an integer, we can rewrite the equation as a + c = 2(k1 + k2 - b). Thus, a + c is even, and the relation is transitive.
Since the relation ∼ satisfies the properties of reflexivity, symmetry, and transitivity, we can conclude that it defines an equivalence relation on Z.
(b) The partition of Z that arises from the equivalence relation ∼ consists of two subsets: the set of even integers and the set of odd integers. Each element in Z belongs to one and only one of these subsets. The even integers are those that can be expressed as 2k, where k is an integer, while the odd integers are those that can be expressed as 2k + 1, where k is an integer. Therefore, the partition can be represented as follows:
Z = {even integers} ⋃ {odd integers}
Each subset in the partition contains elements that are related to each other under the ∼ relation. Elements within the same subset have an even sum when added, while elements from different subsets have an odd sum when added.