   Chapter 8.3, Problem 22ES ### 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 19-31, (1) prove that the relation is an equivalence relation, and (2) describe the distinct equivalence classes of each relation.Let A be the set of all statement forms in three variables p,q, and r. R is relation defined on A as follows: For all P and Q in A, P RQ                               ⇔                       P   and  Q  have the same truth table .

To determine

To prove:

(1) The given relation is an equivalence relation,

(2) Describe the distinct equivalence classes of each relation.

Explanation

Given information:

Let A be the set of all statement forms in three variables p, q, and r.

R Is the relation defined on A as follows: For all P and Q in A,

P R Q ⇔ P and Q have the same truth table.

Proof:

A=Set of all statement forms using the variables p,q and rE={(p,q)A×A|P and Q have the same truth table}

(1).To prove: R Is an equivalent relation

A relation will be equivalence relation if it is reflexive, symmetric and transitive.

Reflexive:

Let xA

Since a statement is always logically equivalent with itself, a statement always has the same truth table as itself and thus (x,x)R.

Since (x,x)R for all xA and thus R is indeed reflexive

Symmetric:

Let us assume that (x,y)R. by the definition of R:

x and y have the same truth table

However, if x and y have the same truth table, then y and x also have the same truth table

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