Discrete Mathematics | Propositional Logic | Test 2 | Question: 4

Question

Consider a conditional statement $P \rightarrow Q.$ The proposition $Q \rightarrow P$ is called the $\textit{converse}$ of $P \rightarrow Q.$ The proposition $\neg Q \rightarrow \neg P$ is called the $\textit{contrapositive}$ of $P \rightarrow Q$ and $\neg P \rightarrow \neg Q$ is called the $\textit{inverse}$ of $P \rightarrow Q.$     
Which of the following statement(s) is/are correct:   
    

• A. If the statement is "if P then Q". Now, its converse is "if Q then P".     
        
        
• B. If the statement is "if P then Q". Now, its inverse is "if not P then not Q".     
          
• C. If the statement is "if P then Q". Now, its contrapositive is "if not Q then not P".      
         
• D. If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.    
  (P and Q are distinct atomic sentences)

Answer 1

• A. If the statement is “If P then Q”. Now, its converse is “if Q then P”.

Statement: “If P then Q” read as $P → Q$

                     “If Q then P” read as $Q → P$, which is $converse$ of the statement.

• A. If the statement is “If P then Q”. Now, its inverse is “If not P then not Q”. 

Statement: “If P then Q” read as $P→ Q$

                    “If not P then not Q” read as $\neg P → \neg Q$, which is $inverse$ of the statement.

100. If the statement is “If P then Q”. Now, its contrapositive is “If not Q then not P”. 

Statement: “If P then Q” read as $P→ Q$

                    “If not Q then not P” read as $\neg Q → \neg P$, which is $contrapositive$ of the statement.

500. If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.    
     (P and Q are distinct atomic sentences) 

$Yes$;

$Conditional$ $Statement$ $\equiv$ $its$ $Contrapositive$.

$Converse$ $Conditional$ $Statement$ $\equiv$ $Inverse$ $of$ $the$ $statement$.

$Ans: A;B;C;D$ 

Comments

“If the statement is true, then the contrapositive is also logically true. If the converse is true, then the inverse is also logically true.”

why ?

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P and Q are distinct atomic sentences. We have $4-cases:$

$P = Q = T; P = T, Q= F; P = F, Q= T; P=Q=F$

$P\to Q  = \neg P \vee Q;   \neg Q\to \neg P = Q \vee \neg P$

$P$	$Q$	$P \to Q$	$\neg Q \to \neg P$
$T$	$T$	$T$	$T$
$T$	$F$	$F$	$F$
$F$	$T$	$T$	$T$
$F$	$F$	$T$	$T$

Both have same truth values; so, $conditional$ $statement \equiv contrapositive$

$Q \to P = \neg Q \vee P;  \neg P \to \neg Q = P \vee\neg Q$

$P$	$Q$	$Q \to P$	$\neg P \to \neg Q$
$T$	$T$	$T$	$T$
$T$	$F$	$T$	$T$
$F$	$T$	$F$	$F$
$F$	$F$	$T$	$T$

Both have same truth values; so, $converse\equiv inverse$