Question

1. A function + : S × S → S for a set S is said to provide an associative binary operation on S if r + (s + t) = (r + s) +t for all r, s, t ∈ S. Show that any associative binary operation + on a set S can have at most one “unit” element, i.e. an element u ∈ S such that (*) s + u = s = u + s for all s ∈ S. (So any vector space contains a unique zero vector.) Hint: Prove that any two “unit” elements, satisfying the characteristic feature (*) must be the same element.

3. Determine whether the given subset S of a vector space V is a subspace by indicating both whether S is closed under + and whether S is closed under ·: (1) {(x, y, z) : y ≤ 0} ⊂ R 3 , (2) {(x, y, z) : x = 0 or z = 0} ⊂ R 3 , (3) R 2 ⊂ C 2 with the underlying field F = C, (4) R 2 ⊂ C 2 where the vector space C 2 over C is viewed as a vector space over R (in what way? explain) and hence the underlying field is taken to be F = R.

Answer 1