The image contains a set of logical derivation exercises (Exercise 10) from a formal logic course or text. The task is to write out derivations to prove each of the given sequents, which involve formulas in propositional logic using symbols like implication (→), conjunction (∧), and variables such as ϕ, ψ, θ, χ.

Exercise 10: Derivations for the following sequents

1. ⊢ (ϕ → (ψ → ψ))
   This is likely a simple application of the identity property in propositional logic.

2. ⊢ ((θ → (ϕ → ψ)) → (θ → (ϕ → ψ)))
   This is a tautology; you need to show that the statement is derivable from itself.

3. {(ϕ → ψ), (ϕ → χ)} ⊢ (ϕ → (ψ ∧ χ))
   You are given the premises $ϕ → ψ$ and $ϕ → χ$, and asked to show that $ϕ → (ψ ∧ χ)$ is derivable.

4. {(ϕ → ψ), ((ϕ ∧ ψ) → χ)} ⊢ (ϕ → χ)
   Starting from the assumptions $ϕ → ψ$ and $(ϕ ∧ ψ) → χ$, you need to derive $ϕ → χ$.

5. {(ϕ → (ψ → χ))} ⊢ ((ϕ ∧ ψ) → χ)
   This involves deriving from an implication within an implication and proving that a conjunction leads to the same conclusion.

6. ⊢ ((θ → ψ) → ((ψ → θ) → θ))
   This is more complex, likely involving contraposition and transitivity in some way.

7. ⊢ ((θ → (ϕ ∧ θ)) → ((ϕ → θ) ∧ (θ → ϕ)))
   A derivation where you work with conjunctions and implications together.

How would you like to approach the derivations for these sequents? I can help with a step-by-step derivation for each one.

Here are some follow-up questions to guide you:

1. Would you like to start with a specific exercise from the list for a detailed proof?
2. Are you familiar with the rules of inference, such as Modus Ponens or Conjunction Introduction, which are often used in these proofs?
3. Do you want a brief refresher on natural deduction or sequent calculus for solving these?
4. Would you prefer formal derivations in symbolic notation or a more intuitive explanation?
5. Are there any specific logical symbols or rules that you need clarification on?

Tip: In propositional logic, common proof strategies include using tautologies, implications, and simplifications like Modus Ponens (if $p → q$ and $p$, then $q$).