Examples of Proofwriting for Math 13 Discussion Problems

Problem 1(a). The domain of f is N × N, so elements of the domain of f are ordered pairs 〈a, b〉 where a, b ∈ N. The following equivalences hold for every ordered pair 〈a, b〉 ∈ N× N. 〈a, b〉 ∈ f[E] ⇐⇒ f(〈a, b〉) ∈ E ⇐⇒ a ∈ E The first equivalence is an immediate consequence of the definition of the inverse image f[E]. The second equivalence follows from the definition of function f . We thus have f[E] = {〈a, b〉 ∈ N× N | a ∈ E} = {〈a, b〉 ∈ N× N | a is even} This can also be equivalently expressed as follows: f[E] = E × N, because for every ordered pair 〈a, b〉, 〈a, b〉 ∈ E × N ⇐⇒ a ∈ E ∧ b ∈ N ⇐⇒ (a ∈ N ∧ b ∈ N) ∧ a ∈ E ⇐⇒ 〈a, b〉 ∈ N× N ∧ a ∈ E. ⊓⊔