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.
xRythe values of x and y are stored in the same memory location during execution of the program.

To prove:

(1) The given relation is an equivalence relation,

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

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 x∈A

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

Since (x,x)∈R for all x∈A,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)∈R, R is symmetric