Gfyuki

8 $begingroup$ Let $mathcal{X}$ be the set of all nonempty subsets of the set ${1,2,3,...,10}$ . Define the relation $mathcal{R}$ on $mathcal{X}$ by: $forall A, B in mathcal{X}, A mathcal{R} B$ iff the smallest element of $A$ is equal to the smallest element of $B$ . For example, ${1,2,3} mathcal{R} {1,3,5,8}$ because the smallest element of ${1,2,3}$ is $1$ which is also the smallest element of ${1,3,5,8}$ . Prove that $mathcal{R}$ is an equivalence relation on $mathcal{X}$ . From my understanding, the definition of reflexive is: $$mathcal{R} text{ is reflexive iff } forall x in mathcal{X}, x mathcal{R} x$$ However, for this problem, you can have the relation with these two sets: ${1}$ and ${1,2}$ Then wouldn't this not be reflexive since $2$ is not in the first set, but is in the second...