Global and Local Minimum

Question

Suppose that we have an objective function X ⊆ R^n  → R and Y ⊂ X.

Which of the following are correct:

1. Every global minimizer of f in X is global minimizer in Y
2. Every local minimizer of f in Y is local minimizer in X

My answer is that 1) and 2) are correct. I want to check if my answers are correct or we can prove something other with some example.