List of relations, union and intersections of relations

Question

I would like to know if these exercises about relations are right. I will appreciate any help.

1. Write a list of all relations from the set $A$ to the set $B$ where $A=\{\alpha,\beta\}$ and $B=\{x\}$.

$R=\{(\alpha,x),(\beta,x)\}$

2. Given the relations $R_1=\{(1,2),(3,4),(5,6),(7,8),(9,10)\}$ and $R_2=\{(a,b),(1,2),(c,d),(3,4)\}$. Find the relation $R_1∩R_2$ and $R_1∪R_2$.

$R_1∩R_2=\{(1,2),(3,4)\}$

$R_1∪R_2=\{(1,2),(3,4),(5,6),(7,8),(9,10),(a,b),(c,d)\}$

Answer 1

Your answer to the second question is correct.

For the first one, recall that a relation from $A$ to $B$ is a subset of $A \times B$. Since in your case, $A$ has two elements and $B$ has one element, the set $A \times B$ has two elements, and the set of subsets of $A \times B$ has $2^2 = 4$ elements. Thus you should find four relations, but you just found one of them...