PineWiki
1. A big union
For each i∈ℕ, define
Ai = { j∈ℕ | j < i }
Define
B = { Ai | i∈ℕ }
What is ∪B, the union of all elements of B? Prove your answer.
2. A big sum
Let f(n) = 0⋅1 + 1⋅2 + 2⋅3 + 3⋅4 + ... + n(n+1). Prove that f(n) = n(n+1)(n+2)/3 for all n in ℕ.
3. Functions
Let f:A→B and g:B→C.
- Prove or disprove: if f is bijective, and g is bijective, then their composition g∘f is bijective.
- Prove or disprove: if g∘f is bijective, then f and g are both bijective.
4. Cancellation
Let F be the set of all functions from ℕ to ℕ. A function f in F has the left cancellation property if
- f∘g = f∘h ⇒ g = h
for all g, h in F, where two functions g and h are equal if and only if g(x) = h(x) for all x in their common domain.
Show that f has the left cancellation property if and only if f is injective.
