To solve this problem, let's break it into two parts: proving transitivity of the relations and showing the additional statement.为了解决这个问题，让我们将其分为两部分：证明关系的传递性和显示附加语句。
Definitions:定义：
- The symbol
≿ represents a preference relation, meaning "is at least as preferred as."符号≿代表偏好关系，意思是“至少与以下一样优先”。 - The symbol
≻ represents strict preference, meaning "is strictly preferred to."符号≻代表严格偏好，意思是“严格优先于”。 - The symbol
∼ represents indifference, meaning "is equally preferred to."符号∼代表冷漠，意思是“同样优先于”。
Part 1: Proving that ≻ is transitive第 1 部分：证明≻是传递性的
We need to show that if x1≻x2 and x2≻x3, then x1≻x3.我们需要证明如果 x1≻x2 和 x2≻x3 ， 然后 x1≻x3 。
Proof:证明：
By definition, x1≻x2 means x1 is strictly preferred to x2, which implies x1 is at least as preferred as x2 (i.e., x1≻x2⇒x1≽x2) and they are not indifferent.根据定义， x1≻x2 方法 x1 严格优先于 x2 ，这意味着 x1 至少与 x2 （IE， x1≻x2⇒x1≽x2 ）而且他们并非无动于衷。
Similarly, x2≻x3 implies x2≽x3 and they are not indifferent.相似地， x2≻x3 意味着 x2≽x3 他们并非无动于衷。
Transitivity of the preference relation: Since ≽ is transitive (as it is a preference relation), from x1≽x2 and x2≽x3, it follows that x1≽x3.偏好关系的传递性：因为 ≽ 是传递性的（因为它是偏好关系），从 x1≽x2 和 x2≽x3 ，由此可知 x1≽x3 。
Strict preference: Additionally, since x1≻x2 and x2≻x3, we know that neither x1∼x2 nor x2∼x3. This implies x1=x3, and thus x1≻x3.严格偏好：另外，因为 x1≻x2 和 x2≻x3 ，我们知道两者都不是 x1∼x2 也不 x2∼x3 。这意味着 x1=x3 ，因此 x1≻x3 。
Therefore, strict preference ≻ is transitive.因此，严格偏好≻是传递的。
Part 2: Proving that ∼ is transitive第 2 部分：证明∼是传递性的
We need to show that if x1∼x2 and x2∼x3, then x1∼x3.我们需要证明如果 x1∼x2 和 x2∼x3 ， 然后 x1∼x3 。
Proof:证明：
By definition, x1∼x2 means x1 is equally preferred to x2, which implies both x1≽x2 and x2≽x1.根据定义， x1∼x2 方法 x1 同样优选 x2 ，这意味着两者 x1≽x2 和 x2≽x1 。
Similarly, x2∼x3 implies x2≽x3 and x3≽x2.相似地， x2∼x3 意味着 x2≽x3 和 x3≽x2 。
By transitivity of ≽, from x1≽x2 and x2≽x3, it follows that x1≽x3.通过传递性 ≽ ， 从 x1≽x2 和 x2≽x3 ，由此可知 x1≽x3 。
Similarly, by transitivity, x3≽x2 and x2≽x1 imply x3≽x1.类似地，根据传递性， x3≽x2 和 x2≽x1 意味着 x3≽x1 。
Since we have both x1≽x3 and x3≽x1, it follows that x1∼x3.既然我们都有 x1≽x3 和 x3≽x1 ，由此可知 x1∼x3 。
Therefore, indifference ∼ is transitive.因此，冷漠∼是传递性的。
Part 3: Proving x1∼x2∼x3 implies x1∼x3第三部分：证明 x1∼x2∼x3 意味着 x1∼x3
This is directly shown by the transitivity of indifference ∼. If x1∼x2 and x2∼x3, then from the transitivity of ∼, we conclude x1∼x3.这直接由冷漠∼的传递性表现出来。如果 x1∼x2 和 x2∼x3 ，那么根据传递性 ∼ ，我们得出结论 x1∼x3 。
Conclusion:结论：
- We have shown that if ≽ is a preference relation, the strict preference
≻ is transitive and indifference ∼ is transitive.我们已经证明，如果 ≽ 是偏好关系，严格偏好≻是传递性的，冷漠∼是传递性的。 - Additionally, if x1∼x2∼x3, then x1∼x3.另外，如果 x1∼x2∼x3 ， 然后 x1∼x3 。