   Chapter 8.2, Problem 11ES ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193

#### Solutions

Chapter
Section ### Discrete Mathematics With Applicat...

5th Edition
EPP + 1 other
ISBN: 9781337694193
Textbook Problem
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# In 9—33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answers.11. D is the relation defined on R as follows: For every x ,   y ∈ R ,   x   D   y ⇔ x y ≥ 0 .

To determine

To determine whether the given relation is reflexive, symmetric, transitive, or none of these.

Explanation

Given information:

D is the relation defined on R as follows:

For all x, y ∈ R, x D y ⇔ xy0.

Calculation:

Let us consider the given relation D as

D={x,yR|xy0}

Reflexive:

The relation D is reflexive if (a,a)D for every element aA.

D is reflexive if it contains (x,x) for all xR.

We note that D contains (x,x) as xx=x20 for all xR (as the square of a real number is always nonnegative) and thus D is reflexive.

Hence, reflexive.

Symmetric:

The relation D on a set A is symmetric if (b,a)D whenever (a,b)D for every elements a,bR.

Let us assume that (a,b)D. by definition of D :

ab0

By commutative property of multiplication:

ba0

This last inequality then implies that (b,a)D and thus D is symmetric

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