Final answer:
Without specific definitions of the equivalence relations ∼1 and ∼2, it is impossible to confirm whether they satisfy reflexivity, symmetry, and transitivity. However, a hint in the text suggests a possible failure at the origin, which could indicate a lack of reflexivity there.
Step-by-step explanation:
The question appears to focus on two equivalence relations, denoted as ∼1 and ∼2, defined on a set X, and our goal is to determine if these relations satisfy the properties of reflexivity, symmetry, and transitivity. These three properties are critical to defining an equivalence relation:
Reflexivity: For all a in X, a ∼ a.Symmetry: For all a, b in X, if a ∼ b, then b ∼ a.Transitivity: For all a, b, and c in X, if a ∼ b and b ∼ c, then a ∼ c.
The text provided above does not contain clear definitions of ∼1 and ∼2 but seems to include references to the relations themselves possibly failing at certain points, such as the origin in Eq. (12.3.5). This could hint at a lack of reflexivity at the origin if the expression is undefined there. An equivalence relation must satisfy all three conditions across the entire set for which they are defined. Therefore, without specific definitions of each relation, it is impossible to determine if they are indeed equivalence relations by verifying reflexivity, symmetry, and transitivity.