   Chapter 8.3, Problem 24ES ### 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.24. Let A be the set of identifiers in a computer program. Itis common for identifiers to be used for only a short part of the execution time of a program and not to be used again to execute other parts of the program. In such cases, arranging for identifiers to share memory locations makes efficient use of a computer’s memory capacity. Define a relation R onA as follows: For all identifiers x and y.xRy the values of x and y are stored in the same memory location during execution of the program.

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 identifiers in a computer program.

Define a relation R on A as follows: For all identifiers x and y ,

x R y ⇔ the values of x and y are stored in the same memory location during execution of the

program

Proof:

A=Set of identifiers in a computer programR={(x,y)A×A|x and y are stored in the same memory location}

(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 identifier x always stored in the same memory location as itself, (x,x)R.

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

Symmetric:

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

x and y are stored in the same memory location

However, if x and y are stored in the same memory location, then y and x are also stored in the same memory location.

(y,x)R

Since (x,y)R implies (y,x)RR is symmetric

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