Questions 5-6 Pre-Write

5. Ambiguity of "formulas"
a. Without the proper parenthesis, it is unclear in which order the formula should be conducted.
((ϕ∧ψ)∨θ) is read as θ and either ϕ or ψ, so either θ ϕ or θ and ψ
(ϕ∧(ψ∨θ)) is read as ϕ or ψ and θ
These are two different answers.
b. Unique readability means that a statement is clear enough that anyone reading it would arrive to the same conclusion.
c. Without a complete WFF it is unclear what operation is being performed.
∀x(x = y) is a complete WFF with ∀x as an initial segment. If a mathematician was given simply ∀x, they would not arrive at the same conclusion as one given the full WFF. Therefore, complete WFF satisfy the conditions for unique readability.

6. AB is rational.
We know that $\sqrt{2}$ is irrational. So, if A=$\sqrt{2}$ and B =$\sqrt{2}$ satisfy the theorem, then we are done. If they do not, then $\sqrt{2}$$\sqrt{2}$ is irrational, so let a be this number. Then, letting B=$\sqrt{2}$, it is easy to verify that aB = 2 which is rational and hence would satisfy the theorem.
($\sqrt{2}$$\sqrt{2}$)$\sqrt{2}$ = $\sqrt{2}$$2$ = 2

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