/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 27 Let \(A=\\{2,3,4,5,6,7,8\\}\) an... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 27

Let \(A=\\{2,3,4,5,6,7,8\\}\) and define a binary relation \(T\) on A as follows: For all \(x, y \in A, \quad x T y \Leftrightarrow 3 \mid(x-y)\).

Short Answer

Expert verified
The binary relation \(T\) on the set \(A = \{2, 3, 4, 5, 6, 7, 8\}\) is defined as \(x T y \Leftrightarrow 3|(x - y)\). After checking all pairs of elements in \(A\), the following pairs satisfy the relation \(T\): (2, 2), (2, 5), (2, 8), (3, 3), (3, 6), (4, 4), (4, 7), (5, 2), (5, 5), (5, 8), (6, 3), (6, 6), (7, 4), (7, 7), (8, 2), (8, 5), and (8, 8).

Step by step solution

01

1. Understand the Set A and the Relation T

First, let's take a closer look at the given set A and the relation T. The set A consists of the integers \(2, 3, 4, 5, 6, 7, 8\), and we can see that T is a binary relation between the elements of A. The relation T is defined as follows: For any two elements x and y in A, x is related to y according to T (denoted as xTy) if and only if 3 divides the difference (x-y).
02

2. Apply the Relation T to the Elements of A

Now, let's apply the relation T to the pairs of elements in A. We need to find all pairs (x, y) that satisfy the condition \(3|(x-y)\). To do this, we can start by finding all differences (x-y) and checking whether 3 divides them. We will compare all pairs of x and y in A: - For x=2: We have: - \(2-2=0\) (3 divides 0, so 2T2) - \(2-3=-1\) (3 does not divide -1) - \(2-4=-2\) (3 does not divide -2) - \(2-5=-3\) (3 divides -3, so 2T5) - \(2-6=-4\) (3 does not divide -4) - \(2-7=-5\) (3 does not divide -5) - \(2-8=-6\) (3 divides -6, so 2T8) - Continue this process for all other elements in A (x=3, 4, 5, 6, 7, 8).
03

3. Determine the Pairs Satisfying the Relation T

After performing step 2, we have checked all pairs of elements in A and can determine which pairs satisfy the relation T. The following pairs satisfy the relation T: - (2, 2), (2, 5), (2, 8) - (3, 3), (3, 6) - (4, 4), (4, 7) - (5, 2), (5, 5), (5, 8) - (6, 3), (6, 6) - (7, 4), (7, 7) - (8, 2), (8, 5), (8, 8) As a result, these are the pairs of elements in A that satisfy the relation T, where the difference (x-y) is divisible by 3.

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