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relational algebra - Set Properties: Irreflexivity and Transitivity

问题描述:

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.

网友答案:

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.

网友答案:

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.

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