In the definition of these relations in the book, the values of variables X and Y in the first-order logical formula are the total individual domain, so there are restrictions of x∈A and y∈A in the domain. In fact, we only consider the set A, so these definitions can completely remove the restrictions of x∈A and Y ∈ A..

When set A is a separate domain, the definition is

(1) If there is x (

(2) If there is X (

(3) If any x any y (

(4) If any x any y (

In this way, it looks simple.

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1, when judging reflexivity and reflexivity, you only look at everything.

2. the relation r on set a is a subset of cartesian product a× a. as long as < x, y > ensures that x, y∈A is enough, x and y don't have to take all the elements in a.

The range in the definition of symmetry and antisymmetry is a hint. For example, in the definition of symmetry, the antecedent of implication is X, Y ∈ A ∧.

So is antisymmetry. Find out from R.