Mathematics
Homomorphism and Isomorphism in Abstract Algebra
Quick fact
Even though the integers and the even integers are different sets, the map f(n) = 2n is an isomorphism from the integers to the even integers (with addition), showing they are structurally identical as groups.
Why this is interesting
Imagine two jigsaw puzzles from different manufacturers. If the pieces fit together in exactly the same patterns, are they the same puzzle even if the pictures differ?