(A) Show That the Relation $$R = \{ ( x , y ) \mid x - y \text { is an even integer } \}$$

question 28

Essay

(a) Show that the relation $$R = \{ ( x , y ) \mid x - y \text { is an even integer } \}$$ is an equivalence relation on the set of real numbers.
(b) What are the equivalence classes of $1 \text { and } \frac { 1 } { 2 } \text { with respect to } R \text { ? }$

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Definitions:

Gini Ratio

A numerical measure of the overall dispersion of income among households, families, or individuals; found graphically by dividing the area between the diagonal line and the Lorenz curve by the entire area below the diagonal line.

Income Inequality

The uneven allocation of income among households or individuals within an economic system.

Lorenz Curve

A graphical representation of the distribution of income or wealth within a society, indicating inequality.

Noncash Transfers

Benefits provided by the government to individuals in forms other than direct cash, such as food stamps or medical aid.