#### Question

(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 identifiers in a computer program. It is 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 on A as follows: For all identifiers x and y,

$x R y \Leftrightarrow$

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

#### Solution

Verified#### Step 1

1 of 5DEFINITIONS

A relation $R$ on a set $A$ is $\textbf{reflexive}$ if $(a,a)\in R$ for every element $a\in A$.

A relation $R$ on a set $A$ is $\textbf{symmetric}$ if $(b,a)\in R$ whenever $(a,b) \in R$

A relation $R$ on a set $A$ is $\textbf{transitive}$ if $(a,b)\in R$ and $(b,c) \in R$ implies $(a,c)\in R$

A relation $R$ is an $\textbf{equivalence relation}$ if the relation $R$ is transitive, symmetric and reflexive.

The $\textbf{equivalence class}$ of $a$ is the set of all elements that are relation to $a$. Notation: $[a]_R$