To find the domain and range of following relations
- {$(p,q) \in P \times P :$ p is a brother of q }, where P is set of all living people
2.{$(x,y) \in R \times R : y^2 = 1- \frac{2}{x^{2}+1} $ }
For 1, i have found out domain to be set of person who has living brother and range is also the same
For 2, for domain i have to find out for what values of $x$, there exists $y \in R$ with given condition. So whatever values of $x$ i put $y^2$ is always $\geq 0$ so i put $y^2 \geq 0$ and solve for $x$. For range is it simply set of all number $\geq 0$.
Is this right? Thanks
For (1) as you say $q$ can be any living person with a living brother, but I would have thought $p$ could be any living male person with a living brother or sister
For (2) $x^2$ must be non negative so $1+x^2 \ge 1$ so $0 \lt \frac{1}{1+x^2} \le 1$ so $-2 \le -\frac{2}{1+x^2} \lt 0$ so $-1 \le 1-\frac{2}{1+x^2} \le 1$ but $y^2$ must be also be non negative so $0 \le y^2 \lt 1$ so $0 \le 1-\frac{2}{1+x^2} \le 1$ so $-1 \le -\frac{2}{1+x^2} \lt 0$ so $0 \lt \frac{1}{1+x^2} \le \frac12$ so $2 \le 1+x^2$ so $1 \le x^2$
Given $1 \le x^2$ and $0 \le y^2 \lt 1$, we can say that $x \le -1$ or $x \ge 1$, and that $-1 \lt y \lt 1$