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Chapter 9: Problem 18

Let \(A=\\{3,4,5\\}\) and \(B=\\{4,5,6\\}\) and let \(S\) be the "divides" relation. That is, For all \((x, y) \in A \times B, \quad x S y \Leftrightarrow x \mid y .\) State explicitly which ordered pairs are in \(S\) and \(S^{-1}\).

Short Answer

Expert verified
The ordered pairs in relation S are: \(S = \{ (3,6), (4,4), (5,5) \}\), while the ordered pairs in the inverse relation S鈦宦� are: \(S^{-1} = \{ (6,3), (4,4), (5,5) \}\).

Step by step solution

01

Find A 脳 B

First, let's find all ordered pairs formed by elements of the sets A and B. This step involves calculating the Cartesian product A 脳 B, which contains all possible ordered pairs with the first element from A and the second element from B: \(A \times B = \{ (3,4), (3,5), (3,6), (4,4), (4,5), (4,6), (5,4), (5,5), (5,6) \}\)
02

Find ordered pairs in S

Now we are looking for the ordered pairs in S by checking if x divides y for each ordered pair (x, y) 鈭� A 脳 B. Recall that x divides y if and only if there exists an integer k such that x * k = y. Let's check for all pairs: (3,4): 3 does not divide 4. (3,5): 3 does not divide 5. (3,6): 3 divides 6, so include (3,6). (4,4): 4 divides 4, so include (4,4). (4,5): 4 does not divide 5. (4,6): 4 does not divide 6. (5,4): 5 does not divide 4. (5,5): 5 divides 5, so include (5,5). (5,6): 5 does not divide 6. So the relation S contains the following ordered pairs: \[S = \{ (3,6), (4,4), (5,5) \}\]
03

Find ordered pairs in S鈦宦�

Now we'd like to find the inverse relation S鈦宦�, which is the set of all ordered pairs where the first and second elements are switched for each ordered pair in S. Specifically, if (x, y) is in S, then (y, x) should be in S鈦宦�. Hence, match the elements from S: (3,6) becomes (6,3) (4,4) stays as (4,4) (5,5) stays as (5,5) So the inverse relation S鈦宦� is as follows: \[S^{-1} = \{ (6,3), (4,4), (5,5) \}\]

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