For any two functions f and g defined as f (x, y )= (x+1,y+3) and g (x,y)=(x-3,y-2) show that the following statement is true or false.

Question

Let $f$ and $g$ be two functions defined on $ \mathbb {R^2} $ as follows $f(x,y)=(x+1,y+3)$ and $g(x,y)=(x-3,y-2)$.

Is the following statement true or false? There exists $(a,b)$ in $ \mathbb {R^2} $ such that $f(a,b) \neq g(x,y)$ for any $(x,y)$.

The answer is "false". But I think the statement is true. Because for any $(x,y)$ there will be an $(a,b)$ (in fact more than one) such that $f(a,b) \neq g(x,y)$. Am I right?

Answer 1

Fix some arbitrary pair $(a,b)$. Then $f(a,b) = (a+1,b+3)$. Put $(x,y) = (a+4, b+5)$. Then $g(x,y) = (x-3,y-2) = (a+4-3, b+5-2) = (a+1,b+3)$, which is $f(a,b)$.

Thus, no matter what pair $(a,b)$ you choose, there always exists a pair $(x,y)$, such that $g(x,y) = f(a,b)$. This means that the statement is indeed false.