Call a logical proposition truth-preserving if the proposition is true under the all-true truth assignment. That is, a proposition is truth-preserving if and only if the first row of its truth table...

Call a logical proposition truth-preserving if the proposition is true under the all-true truth assignment. That is, a proposition is truth-preserving if and only if the first row of its truth table is True.) Prove the following claim by structural induction on the form of the proposition:

Any logical proposition that uses only the logical connectives ∨ and ∧ is truth-preserving. (A solution to this exercise yields a rigorous solution to

Exercise 4.71—there are propositions that cannot be expressed using only ∧ and ∨. Explain.)

Exercise 4.71

rove that the set {∧,∨} is not universal. (Hint: what happens under the all-true truth assignment?)