Discrete Mathematics | Propositional Logic | Test 2 | Question: 3

Question

Suppose we have to construct a formula that expresses the truth function $\phi$ given by:       
$$
\begin{array}{c|c|c}
p & q & \phi \\\hline
T & T & T \\
T & F & T \\
F & T & F \\
F & F & F
\end{array}
$$
(T stands for True and F stands for False)      
Which of the following statement(s) is/are correct ?   

• A. The formula that $\phi$ expresses is $p$       
• B. The formula that $\phi$ expresses is $(p \wedge q) \vee (p \wedge \neg q)$        
• C. The formula that $\phi$ expresses is $(p \wedge \neg r \wedge q) \vee (~q \wedge p) \vee (r \wedge p) \vee p$      
• D. The formula that $\phi$ expresses is $\neg p \rightarrow ((p \rightarrow \neg p) \rightarrow (\neg p \rightarrow p))$

Answer 1

• A. From truth table we can see that $\phi$ = p because columns of p & $\phi$ are same.
• B. (pq) + (pq’) = p(q+q’) = p
• C. (pr’q) + (qp) + (rp) + (p) = (r’q + q + r + 1)p = p
• D. p’ → ((p → p’) → (p’ → p)) = p + ((p → p’) → (p’ → p)) = p + ((p’ + p’) → (p + p)) = p + (p’ → p) = p + (p + p) = p.

Ans is A,B,C,D.