I would like to know If these exercises about relations are right. I took discrete mathematics like 10 years ago, but I don't remember taking relations. I'm using a book I found online but does not have examples like these ones. I will appreciate any help.
1. Find the relation $M$ over a set $S=\{1,2,3\}$, if $M=\{(x,r(x)):r(x)=2x-1\}$
I substituted the values in the set $S$ in the equation in the set $M$.
- For $x=1$, I got $r(x)=1$
- For $x=2$, I got $r(x)=3$
- For $x=3$, I got $r(x)=5$
So $M=\{(1,1),(2,3),(3,5)\}$
2. Find the set of coordinate pairs $\{(x,y)\}$ si $y=x^2-2x-3$ and $D=\{x|x\in Z, 1 \leq x\leq4\}$.
I substituted the values in the set $D$ in the equation.
- For $x=1$, I got $y=-4$
- For $x=2$, I got $y=-3$
- For $x=3$, I got $y=0$
- For $x=4$, I got $y=5$
The set of coordinate pairs is: $\{(1,-4),(2,-3),(3,0),(4,5)\}$
3. Find the relation $Q$ over $S \times T$ if $S=\{1,2,3\}$, $T=\{4,5\}$, and the correspondence rule is $r(x)=x+2$.
$S \times T = \{(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)\}$
So $Q=\{(2,4),(3,5)\}$
Thanks.