   Chapter 8.2, Problem 27ES ### 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 answer. Let A be the set of all English statements. A relation I is defined on A as follows: For every p , q ∈ A , p | q ⇔ p → q   is true .

To determine

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

Explanation

Given information:

Let A be the set of all English statements. A relation I Is defined on A as follows:

For all p, q ∈ A, p I q ⇔ p → q is true.

Calculation:

A= Set of all english statements

I={p,qA|pq is true}

Reflexive:

The relation I Is reflective if (p,p)I for every element pA.

Let p be an English statement.

If p is true, then pp is true, since both (sub) proportions are true.

If p is false, then pp is true, since the if-statement is false.

Thus pp is always rue, which implies that (p,p)I and thus I Is reflexive.

Symmetric:

The relation I on a set A is symmetric if (p,q)I whenever (q,p)I

Let us assume that (p,q)I, by definition of I:

pq is true

However, when p is false and q is true, then qp will be false and thus qp is not necessarily true, which implies that I Is not symmetric

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