Question

Define a relation T from R to R as follows: For all real numbers x and y

(X,y) E T means that y^2- x^2= 1.

Is T a function? Explain

Answer 1

The relation T: A->B is called a function if and only if $\forall a\in A \exists! b\in B: f(a)=b$

In the given case we have T:R->R: (x, y) $\isin$ T$\iff y^2-x^2=1$

$y^2-x^2=1\implies y=±\sqrt{x^2+1}$

As we can see, for x = 0 y can be both 1 and -1, so the condition of the function doesn't met. T is not a function