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thrown into an indefinite number of equivalent forms. For example, the relation ${\displaystyle x+y=1}$ is equivalent to the relations

${\displaystyle y+x=1}$, ${\displaystyle (x-y)+2y=1}$, ${\displaystyle 6x+6y=6}$,

and so on. Thus a skilful mathematician uses that equivalent form of the relation under consideration which is most convenient for his immediate purpose.

It is not in general true that, when a pair of terms satisfy some fixed relation, if one of the terms is given the other is also definitely determined. For example, when ${\displaystyle x}$ and ${\displaystyle y}$ satisfy ${\displaystyle y^{2}=x}$, if ${\displaystyle x=4}$, ${\displaystyle y}$ can be ${\displaystyle \pm 2}$, thus, for any positive value of ${\displaystyle x}$ there are alternative values for ${\displaystyle y}$. Also in the relation ${\displaystyle x+y>1}$, when either a or ${\displaystyle y}$ is given, an indefinite number of values remain open for the other.

Again there is another important point to be noticed. If we restrict ourselves to positive numbers, integral or fractional, in considering the relation ${\displaystyle x+y=1}$, then, if either ${\displaystyle x}$ or be greater than ${\displaystyle 1}$, there is no positive number which the other can assume so as to satisfy the relation. Thus the "field" of the relation for ${\displaystyle x}$ is restricted to numbers less than ${\displaystyle 1}$, and similarly for the "field" open to ${\displaystyle y}$. Again, consider integral numbers only, positive or negative, and take the relation