Let S be the set of all three-digit numbers, and define x~y to mean that x and y have the same first and last digit. (i) Show that the relations ~ is transitive. (ii) List two numbers in the equivalence class [737] and two numbers not in this equivalence class. (justify with brief explanation).
Alright, so that is the question I am confused by part i, I know that for something to be transitive aRb brc arc all have to be true, but I just don't understand how I am supposed to explain that with the information given. Also for part ii I would assume [747] to be in the same equivalence class and [676] to not be because the first and last digits are the same and then for the second one the first and last digits are now 6 making it a different class. Am I correct with or am I totally of?
Choose any $3$ three-digit numbers $ABC, DEF, GHI \in S$ such that $ABC \sim DEF$ and $DEF \sim GHI$ (here, I am treating $A, B, C, D, E, F, G, H, I$ as individual digits). Then $A = D$ and $C = F$, and similarly $D = G$ and $F = I$. Hence, since $A = G$ and $C = I$, it follows that $ABC \sim GHI$, as desired.
You are correct for part (b).