1. Consider the set of real numbers ℝ. Say that x ≡ y (mod 2π) if x − y is an integer multiple of 2π. Verify that this is an equivalence relation.
2. Put a relation on C[0, 1] by f ∼ g if f(k/10) = g(k/10) for 0 ≤ k ≤ 10.
- (a) Verify that this is an equivalence relation.
- (b) Describe the equivalence classes.
- (c) Show that [f] + [g] = [f + g] is a well-defined operation.
- (d) Show that t[f] = [tf] is well defined for all t ∈ ℝ and f ∈ C[0, 1].
- (e) Show that these operations make C[0, 1]/∼ into a vector space of dimension 11.
3. Say that two real vector spaces V and W are isomorphic if there is an invertible linear map T of V onto W.
- (a) Prove that this is an equivalence relation on the collection of all vector spaces.
- (b) When are two finite-dimensional vector spaces isomorphic?
4. Define |x| = max{x, −x}.
- (a) Prove that |xy| = |x| |y| and |x−1| = |x|−1.
- (b) Prove the triangle inequality |x + y| ≤ |x| + |y|.
5. Prove that if x < y, then there is a rational number r with a finite decimal expansion such that x < r < y.
6. Prove that if x < y, then there is an irrational number z such that x < z < y.