集合属性：非自反性和传递性
这不是作业，但和我的作业有直接关系。换句话说，我需要知道这些信息才能完成我的作业。
R 是否具有传递性：R = {(a,b),(b,a),(c,c)}？我认为它还需要包括 (a,a),(b,b) 但我不确定。
空集 {} 是非自反的吗？
这些情况尚未得到明确解释，我希望得到澄清。
This is not homework but has a direct relation to my homework. In other words, I need to know this information to be able to do my homework.
Is R transitive: R = {(a,b),(b,a),(c,c)}? I would think that it would also need to include (a,a),(b,b) but I am unsure.
Is the empty set {} irreflexive?
These are cases which have not been explained clearly and I would appreciate clarification.
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例如，如果您查看维基百科：传递关系，您就会得到这个很好的量化表达式，如果您的关系是传递性的。
因为它是普遍量化的，所以对于空集来说是正确的（因为关于空集的普遍量化表达式根据定义是正确的）。
你是绝对正确的。如果 R 中有
(a,b)和(b,a)，那么也必须有(a,a)R 是传递性的。非自反性也被普遍量化（“它是一个集合上的二元关系，其中没有元素与其自身相关。” =>
∀x:~(xRx)或~∃x:xRx)，因此它适用于空集。If you look for example at Wikipedia: Transitive relation you have this nice quantified expression that becomes true if your relation is transitive.
Because it's universally quantified it's correct for the empty set (because universally quantified expressions about the empty set are true by definition).
And you are absolutely right. If there is
(a,b)and(b,a)in R, then there also has to be(a,a)for R to be transitive.The irreflexivity is also universally quantified ("It is a binary relation on a set where no element is related to itself." =>
∀x:~(xRx)or~∃x:xRx), so it holds for the empty set.传递律，在数学和逻辑中，是指如果 A 与 B 具有某种关系，并且 B 与 C 具有相同的关系，则 A 与 C 具有相同的关系。在算术中，等式的性质是传递的，因为如果 A = B 且 B如果两个不等式具有相同的含义，则财产不等式也是如此：即，如果 A 大于 B（即 A > B）且 B > B。 C，则A＞ C;如果A小于B(即A＜B)并且B＜B，则A＜B。 C，则A＜0 C. 不及物关系的一个例子是：如果 B 是 A 的女儿，C 是 B 的女儿，那么 C 就不是 A 的女儿；以及非传递关系：如果 A 爱 B，B 爱 C，那么 A 可能爱也可能不爱 C。
非自反关系或反自反关系与自反关系相反。它是集合上的二元关系，其中没有元素与其自身相关。一个例子是“大于”关系(x＞y)。请注意，并非所有非自反关系都是非自反关系。可以定义某些元素与其自身相关但与其他元素无关的关系。例如，二元关系“x和y的乘积是偶数”在偶数集合上是自反的，在奇数集合上是非自反的，在自然数集合上都不是。
Transitive law, in mathematics and logic, statement that if A bears some relation to B and B bears the same relation to C, then A bears it to C. In arithmetic, the property of equality is transitive, for if A = B and B = C, then A = C. Likewise is the property inequality if the two inequalities have the same sense: that is, if A is greater than B (i.e., A > B) and B > C, then A > C; and if A is less than B (i.e., A < B) and B < C, then A < C. An example of an intransitive relation is: if B is the daughter of A, and C is the daughter of B, then C is not the daughter of A; and of a nontransitive relation: if A loves B, and B loves C, then A may or may not love C.
An irreflexive, or anti-reflexive, relation is the opposite of a reflexive relation. It is a binary relation on a set where no element is related to itself. An example is the "greater than" relation (x>y). Note that not every relation which is not reflexive is irreflexive; it is possible to define relations where some elements are related to themselves but not others. For example, the binary relation "the product of x and y is even" is reflexive on the set of even numbers, irreflexive on the set of odd numbers, and neither on the set of natural numbers.