Consider the following sets and binary operations, and identify which set is closed under which binary operations.
- natural numbers under addition
- 2x2 nonsingular matrices under matrix addition
- 2x2 nonsingular matrices under matrix multiplication
- 2x2 matrices with determinant 1 under matrix multiplication
- continuous functions between [0,1] and [0,1] under function composition
for each pair, you should clearly state your answer and provide a rigorous explanation.