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Given a set A = {1, 2, 3, 4}. Define a relation R on A as: R = {(x, y) | x - y is even} List the ordered pairs of the relation R. Is the relation R reflexive, symmetric, and transitive? Justify your answers.

Given a set A = {1, 2, 3, 4}. Define a relation R on A as: R = {(x, y) | x - y is even} List the ordered pairs of the relation R. Is the relation R reflexive, symmetric, and transitive? Justify your answers.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Given a set A = {1, 2, 3, 4}. Define a relation R on A as:

R = {(x, y) | x - y is even}

  1. List the ordered pairs of the relation R.
  2. Is the relation R reflexive, symmetric, and transitive? Justify your answers.
Expert Solution
Step 1: Conceptual Introduction

For the relation R to hold between x and y, the difference (x - y) should be even.

An even number is divisible by 2.

The properties of reflexivity, symmetry, and transitivity can be described as:

  • Reflexive: If every element in set A is related to itself, i.e., (a, a) ∈ R for every a in A.

  • Symmetric: If for every (a, b) in R, (b, a) is also in R.

  • Transitive: If for every pair (a, b) and (b, c) in R, the pair (a, c) is also in R.
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