1. (6 分) 判断下列关系是否是重存在的、不存在的、对称的、反对称的

(a) f(a; b) j a 的课程比 b, a 少； b 2Studentsg
(b) f(x; y) j x y > 0; x 2 Zg
(c) f(x; y) j x = 2; X; y 2 Zg

2. (6 分) 令 R = f(2; 1); (3; 1); (2; 2); (1; 3)g 在 f1 上； 2; 3g 和 S = f(1; 2); (3; 3); (2; 1); (1; 3); (4; 1)g

（一）R\S
(b) R S。
(c) R S

3.（6分）给定以下关系，执行每个指定的操作。
P = f(x; y) j x = 2y x; y 2 Zg,
Q = f(x; y) j x = 2y + 1 x; y 2 Zg,
R = f(x; y) j xy > 0 x; y 2 Rg，和
S = f(x; y) j xy < 0 x; y 2 Rg
(一) P\Q
(b) P [ Q
(c) R S

4.（4 分）对于以下每个关系，给出该关系的矩阵表示。然后，

(a) R = f(1; 1); (1; 2); (2; 2); (3; 1); (3; 3)g
(b) R = f(x; y) j x%y = 0; X; y 2 Hg 在集合 H = f1 上； 2; 3; 4克。

5.（6 分）在讲座中我们学习了如何使用矩阵确定关系的属性。我们是

1. (6 Fēn) pànduàn xiàliè guānxì shìfǒu shì zhòng cúnzà

1. (6 points) Determine if each of the following relations are re exive, irre exive, symmetric, antisym-
metric, and/or transitive.
(a) f(a; b) j a is taking fewer courses than b, a; b 2Studentsg
(b) f(x; y) j x  y > 0; x 2 Zg
(c) f(x; y) j x = 2; x; y 2 Zg

2. (6 points) Let R = f(2; 1); (3; 1); (2; 2); (1; 3)g on f1; 2; 3g and S = f(1; 2); (3; 3); (2; 1); (1; 3); (4; 1)g
on f1; 2; 3; 4g. Perform the following operations:
(a) R \ S
(b) R S.
(c) R  S

3. (6 points) Given the following relations, perform each speci ed operation.
P = f(x; y) j x = 2y x; y 2 Zg,
Q = f(x; y) j x = 2y + 1 x; y 2 Zg,
R = f(x; y) j xy > 0 x; y 2 Rg, and
S = f(x; y) j xy < 0 x; y 2 Rg
(a) P \ Q
(b) P [ Q
(c) R  S

4. (4 points) For each of the following relations, give the matrix representation for that relation. Then,
using the matrix, determine if the relation is re exive, symmetric, antisymmetric, and/or transitive.