Mathematics

Problem

Consider the set

\mathbb Z_{5}=\{-5,-4,\dots,0,\dots,4,5\}

of integers from

-5

5

. A relation

T

\mathbb Z_{5}

is defined by

T=\{(x,y)\mid y^{2}=25-x^{2},\;x, y\in\mathbb Z_{5}\}.

(i) For

x=0,1,2

list all corresponding values of

y

that belong to

T

. (ii) Is

T

a function from

\mathbb Z_{5}

\mathbb Z_{5}

? Give a clear reason based on the outputs obtained in part (i) and on the defining rule. (iii) Suggest a restriction on the domain or the range (state it explicitly) that converts

T

into a function, and justify why the restricted relation satisfies the definition of a function.

Intuitive explanation

The relation $T$ is defined by a circle equation $y^2 = 25 - x^2$ , which rearranges to $x^2 + y^2 = 25$ . For each $x$ -value, you need to find all integers $y$ in $\mathbb Z_5$ that satisfy this constraint—there may be zero, one, or multiple solutions. The key insight is that a relation becomes a function only when each input has exactly one output, so you'll need to check whether $T$ passes this test and, if not, how to modify it.

Intuitive explanation

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