Let R ⊆ Nk be a k-ary relation. Say that R is definable in Th(N,+) if we can give a formula ‑ with k free variables x₁, . . . , x_k such that for all a1, . . . , a_k ∈ N, ‑(a₁, . . . , a_k) is true exactly when a₁, . . . , a_k ∈ R. Show that each of the following relations is definable in Th(N,+).

Aa. R₀ = {0}

b. R₁ = {1}

c. R= = {(a, a)| a ∈ N}

d. R_< = {(a, b)| a, b ∈ N and a < b}