[Math] Proof With and Without Truth Tables

logic

$(6)$

Use truth tables to determine whether or not the following argument is correct:
"If the tax rate and the unemployment rate both go up, then there will be a recession.
If the GNP goes up, then there will not be a recession. The GNP and taxes are both going
up. Therefore, the unemployment rate is not going up."
In other words, decide whether the concluding statement must be true, given that the
preceding compound statements are true.

My Work

Let $P :=$ "The tax rate is going up." Let $Q :=$ "The unemployment rate is going up." Let $R :=$ "There is a recession." Let $S :=$ "The GNP is going up." Then "If the tax rate and the unemployment rate both go up, then there will be a recession" has the logical form $(P \wedge Q) \Longrightarrow R$. Likewise, the statement "If the GNP goes up, then there will not be a recession" has the logical format $S \Longrightarrow \sim R$. Using truth tables, we will prove the truth or falsity of the statement "The GNP and taxes are both going up. Therefore, the unemployment rate is not going up" which can be expressed as follows: $(S \wedge P) \Longrightarrow \sim Q$ which, by a previously given equivalence, is equivalent to $\sim (S \wedge P) \vee \sim Q$ .

$(7)$

Do $(6)$ without truth tables.

Proof: (My Work)
From a previous question $(5)$ I have the following equivalences:

Use the following equivalances: $$ P \Longrightarrow Q \equiv \ \sim P \vee Q \qquad \sim \sim P \equiv P \qquad \sim (P \wedge Q) \equiv \sim P \ \vee \sim Q $$ to prove that $P \Longrightarrow Q \vee R \equiv P \wedge \sim Q \Longrightarrow R$.

\begin{align*}
(S \wedge P) \Longrightarrow \sim Q &\equiv \ \sim (S \wedge P) \vee \sim Q \\
\sim (S \wedge P) \vee \sim Q &\equiv \ \sim(S \wedge P \wedge Q) \\
\end{align*}
Since $S \Longrightarrow \sim R$ and $(P \wedge Q) \Longrightarrow R$, $(S \wedge P \wedge Q) \equiv (R \wedge \sim R)$. By the Law of the Excluded Middle, $(R \wedge \sim R)$ is false for all $R$, so $\sim(R \wedge \sim R)$ is true for all $R$, so we have that $(S \wedge P) \Longrightarrow \sim Q$ is a true statement.

My Question

I can do the proof without truth tables, but I am not sure what I should do to prove it with truth tables. Should I construct the truth values of every possible configuration of $P, Q, R, S$ and piece together the following statement:

Therefore, the unemployment rate is not going up" which can be expressed as follows: $(S \wedge P) \Longrightarrow \sim Q$ which, by a previously given equivalence, is equivalent to $\sim (S \wedge P) \vee \sim Q$ .

Best Answer

If you number the statements as

  1. If the tax rate and the unemployment rate both go up, then there will be a recession
  2. If the GNP goes up, then there will not be a recession
  3. The GNP and taxes are both going up

then you can exclude all but one of the rows of the truth table as being contrary to these statements 

P Q R S Comment

T T T T No - 2
T T T F No - 3
T T F T No - 1
T T F F No - 1, 3
T F T T No - 2
T F T F No - 3
T F F T 
T F F F No - 3
F T T T No - 2, 3
F T T F No - 3
F T F T No - 3
F T F F No - 3
F F T T No - 2, 3 
F F T F No - 3
F F F T No - 3
F F F F No - 3

That leaves $P$ and $S$ true and $R$ and $Q$ false. In particular $Q$ is false and so unemployment is not going up.

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