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,

xRyx R y \Leftrightarrow

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

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DEFINITIONS

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

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

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

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

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

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