Suppose that a is a nonempty set, and f is a function that has a as its domain. let r be the relation on a consisting of all ordered pairs (x, y) such that f(x) = f(y). (a) show that r is an equivalence relation on a.

See proof below

Step-by-step explanation:

Let xRy denote the statement "(x,y)∈A". To prove that R is an equivalence relation on A, we must prove that R

For the first one, we know that equality is reflexive, and for all x, f(x) is unique because f is a function, then f(x)=f(x), that is, xRX. So R is reflexive.

For symmetry, suppose that xRy. Then, by definition of R, f(x)=f(y) (again, this value is unique for all x,y). Equality is symmetric, then f(y)=f(x), hence yRx. We have that R is symmetric.

For the last one, suppose that xRy and yRz. Then f(x)=f(y) and f(y)=f(z). f(x),f(y) and f(z) are uniquely determined by the function f. Moreover, equality is transitive, then f(x)=f(y)=f(z) implies that f(x)=f(z), that is, xRz. Hence R is transitive.

R is reflexive, symmetric and transitive, therefore R is a equivalence relation on A.

The hypothesis on f is important. If f wasn't a function, f(x) could represent than one element, so it would be ambiguous and the properties of equality wouldn't necessarily hold.

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step-by-step explanation: