(Sorry for any spelling mistake, English isn't my mothertongue)
Today at school we took a logic exam. One of the questions asked us to find the domain and range of the relations xRy and xSy, where R = y^2 + x^2 < 4 and S = x – 2 < y^2.
Now, I know the first inequality is a circle and the second one is a parabole. I found the domain and range of both binary relations easily by graphing them. However, when I was asked to find the domain and range of the intersection between R and S, I was completely lost. The professor taught us that we can find the intersection by plugging one of the variables of one relation into the other relation if we were working with equations, but is that correct if both are inequalities?
Thanks in advance.
Best Answer
You give us $R=\{(x,y):x^2+y^2<4\}$ and $S=\{(x,y):x-2<y^2\}$
Look at those graphs. Better yet, plot them on the same plane.
$R$ is a disc; the region inside the circle centred on the origin with a radius of 2.
$S$ is the semi-infinite region to the left of a parabola, which has a vertex at $(2,0)$ and pivots around the positive x-axis.
$R$ is thus a proper subset of $S$, and hence the intersection of $R$ with $S$ is exactly $R$.
$$R\cap S=R$$