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这是一篇美国作业案例分享主要为离散数学相关的assignment数学作业
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
公制，和/或传递。
(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
在 f1 上； 2; 3; 4克。执行以下操作：
（一）RS
(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) PQ (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， 使用矩阵，确定关系是否是重复的、对称的、反对称的和/或传递的。 展示你的作品。 (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 还能够使用矩阵来查找两个关系的并、交和复合。这三个 关系上的操作对应于零一的连接、相遇和布尔乘积运算符 矩阵。对于两个关系 A 和 B，我们可以通过在 它们对应的矩阵（MA ^MB），它们的交集通过在它们的 对应的矩阵 (MA _ MB)，以及它们的组合 (A B)，取布尔乘积 它们对应的矩阵（MB MA，注意矩阵的顺序）。令 R 和 S 为 集合 fa 上的关系；乙； cg 对应于以下矩阵。使用下面的矩阵来执行 对关系 R 和 S 进行以下操作。 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 specied 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. Show your work. (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.