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.
(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 on the set H = f1; 2; 3; 4g.

5. (6 points) In lecture we learned how to determine properties of relations using matrices. We are
also able to use matrices to nd the union, intersection and composite of two relations. These three
operations on relations correspond to the join, meet, and boolean product operators of zero-one
matrices. For two relations A and B, we can nd their union by using the \join” operator on
their corresponding matrices (MA ^MB), their intersection by using the \meet” operator on their
corresponding matrices (MA _ MB), and their composite (A  B) by taking the boolean product
of their corresponding matrices (MB MA, note the order of the matrices). Let R and S be the
relations on the set fa; b; cg corresponding to the below matrices. Use the below matrices to perform
the following operations on the relations R and S.

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