Converse, Inverse, and Contrapositive

In mathematical logic and reasoning, conditional statements are important for constructing logical arguments and proofs. A conditional statement is of the form "If p, then q," where p and q are propositions. The statement asserts that if p is true, then q must also be true. However, from a given conditional statement, we can derive three related statements: the converse, the inverse, and the contrapositive.

Converse, Inverse, and Contrapositive

Given a statement of the form: "if $p$ then $q$", then we can create three related statements:

Converse

To form the converse of the conditional statement, interchange $p$ and $q$.

The converse of “If you are born in some country, then you are a citizen of that country” is “If you are a citizen of some country, then you are born in that country.”

Inverse

To form the inverse of the conditional statement, take the negation of both the $p$ and $q$.

The inverse of “If you are born in some country, then you are a citizen of that country” is “If you are not born in some country, then you are not a citizen of that country.”

Contrapositive

To form the contrapositive of the conditional statement, interchange the $p$ and $q$ and take the negation of both.

The Contrapositive of “If you are born in some country, then you are a citizen of that country” is “If you are not a citizen of that country, then you are not born in some country.”

These can be summarized as

$\begin{array}{|c|c|c|}\hline \text {Statement} & \mathrm{\;\;\;}{\text { If } p, \text { then } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}p\rightarrow q \mathrm{\;\;\;}\\ \hline \text { Converse } & \mathrm{\;\;\;}{\text { If } q, \text { then } p} \mathrm{\;\;\;}&\mathrm{\;\;\;}q\rightarrow p \mathrm{\;\;\;} \\ \hline \text { Inverse } & \mathrm{\;\;\;}{\text { If not } p, \text { then not } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim p) \rightarrow(\sim q) \mathrm{\;\;\;} \\ \hline \text { Contrapositive } & \mathrm{\;\;\;}{\text { If not } q, \text { then not } p} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim q) \rightarrow(\sim p) \mathrm{\;\;\;}\\ \hline\end{array}$

Note:

1. A given statement and its contrapositive have the same meaning
2. As the inverse is the contrapositive of the converse, so it has the same meaning as the converse
3. A given statement (= contrapositive) is NOT the same as its converse (=Inverse)

Solved Examples Based on Converse, Inverse, and Contrapositive

Example 1: Contrapositive of the statement ''If two numbers are not equal, then their squares are not equal.'' is:

1) If the squares of two numbers are equal, then the numbers are equal.

2) If the squares of two numbers are not equal, then the numbers are not equal.

3) If the squares of two numbers are equal, then the numbers are not equal.

4) If the squares of two numbers are not equal, then the numbers are equal.

Solution

Converse, Inverse, and Contrapositive -

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

The Contrapositive of “If you are born in some country, then you are a citizen of that country”

is “If you are not a citizen of that country, then you are not born in some country.”

$\begin{array}{|c|c|c|}\hline \text {Statement} & \mathrm{\;\;\;}{\text { If } p, \text { then } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}p\rightarrow q \mathrm{\;\;\;}\\ \hline \text { Converse } & \mathrm{\;\;\;}{\text { If } q, \text { then } p} \mathrm{\;\;\;}&\mathrm{\;\;\;}q\rightarrow p \mathrm{\;\;\;} \\ \hline \text { Inverse } & \mathrm{\;\;\;}{\text { If not } p, \text { then not } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim p) \rightarrow(\sim q) \mathrm{\;\;\;} \\ \hline \text { Contrapositive } & \mathrm{\;\;\;}{\text { If not } q, \text { then not } p} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim q) \rightarrow(\sim p) \mathrm{\;\;\;}\\ \hline\end{array}$

If the squares of two numbers are equal, then the numbers are equal.

Example 2: Find the correct negation of

p : America is not in India

1) p : America is not not in India

2) q : India is in America

3) r : India is not in America

4) s : America is in India

Solution

Converse, Inverse, and Contrapositive -

To form the inverse of the conditional statement, take the negation of both the hypothesis and the conclusion.

The inverse of “If you are born in some country, then you are a citizen of that country”

is “If you are not born in some country, then you are not a citizen of that country.”

$\begin{array}{|c|c|c|}\hline \text {Statement} & \mathrm{\;\;\;}{\text { If } p, \text { then } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}p\rightarrow q \mathrm{\;\;\;}\\ \hline \text { Converse } & \mathrm{\;\;\;}{\text { If } q, \text { then } p} \mathrm{\;\;\;}&\mathrm{\;\;\;}q\rightarrow p \mathrm{\;\;\;} \\ \hline \text { Inverse } & \mathrm{\;\;\;}{\text { If not } p, \text { then not } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim p) \rightarrow(\sim q) \mathrm{\;\;\;} \\ \hline \text { Contrapositive } & \mathrm{\;\;\;}{\text { If not } q, \text { then not } p} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim q) \rightarrow(\sim p) \mathrm{\;\;\;}\\ \hline\end{array}$

America is in India

Example 3: The contrapositive of the following statement, “If the side of a square doubles, then its area increases four times”, is

1) If the side of a square is not doubled, then its area does not increase four times.

2) If the area of a square increases four times, then its side is doubled.

3) If the area of a square increases four times, then its side is not doubled.

4) If the area of a square does not increase four times, then its side is not doubled.

Solution

Converse, Inverse, and Contrapositive -

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

The Contrapositive of “If you are born in some country, then you are a citizen of that country”

is “If you are not a citizen of that country, then you are not born in some country.”

$\begin{array}{|c|c|c|}\hline \text {Statement} & \mathrm{\;\;\;}{\text { If } p, \text { then } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}p\rightarrow q \mathrm{\;\;\;}\\ \hline \text { Converse } & \mathrm{\;\;\;}{\text { If } q, \text { then } p} \mathrm{\;\;\;}&\mathrm{\;\;\;}q\rightarrow p \mathrm{\;\;\;} \\ \hline \text { Inverse } & \mathrm{\;\;\;}{\text { If not } p, \text { then not } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim p) \rightarrow(\sim q) \mathrm{\;\;\;} \\ \hline \text { Contrapositive } & \mathrm{\;\;\;}{\text { If not } q, \text { then not } p} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim q) \rightarrow(\sim p) \mathrm{\;\;\;}\\ \hline\end{array}$

If the statement is $p \rightarrow q$ contrapositive is $\sim q \rightarrow \sim p$

If the area of the square does not increase four times, then its side is not doubled.

Example 4: Let $A, B, C$ and $D$ be four non-empty sets. The contrapositive statement of " If $A \subseteq B$ and $B \subseteq D$, then $A \subseteq C^{\prime \prime}$ is :
1) If $A \subseteq C$, then $B \subset A$ or $D \subset B$
2) If $A \nsubseteq C$, then $A \subseteq B$ and $B \subseteq D$
3) If $A \nsubseteq C$, then $A \nsubseteq B$ and $B \subseteq D$
4) If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \nsubseteq D$

Solution

Converse, Inverse, and Contrapositive -

Given an if-then statement "if $p$ , then $q$ ," we can create three related statements:

A conditional statement consists of two parts, a hypothesis in the “if” clause and a conclusion in the “then” clause. For instance, “If you are born in some country, then you are a citizen of that country”

"you are born in some country" is the hypothesis.

"you are a citizen of that country" is the conclusion.

To form the contrapositive of the conditional statement, interchange the hypothesis and the conclusion of the inverse statement.

The Contrapositive of “If you are born in some country, then you are a citizen of that country”

is “If you are not a citizen of that country, then you are not born in some country.”

$\begin{array}{|c|c|c|}\hline \text {Statement} & \mathrm{\;\;\;}{\text { If } p, \text { then } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}p\rightarrow q \mathrm{\;\;\;}\\ \hline \text { Converse } & \mathrm{\;\;\;}{\text { If } q, \text { then } p} \mathrm{\;\;\;}&\mathrm{\;\;\;}q\rightarrow p \mathrm{\;\;\;} \\ \hline \text { Inverse } & \mathrm{\;\;\;}{\text { If not } p, \text { then not } q} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim p) \rightarrow(\sim q) \mathrm{\;\;\;} \\ \hline \text { Contrapositive } & \mathrm{\;\;\;}{\text { If not } q, \text { then not } p} \mathrm{\;\;\;}& \mathrm{\;\;\;}(\sim q) \rightarrow(\sim p) \mathrm{\;\;\;}\\ \hline\end{array}$

Let $P=A \subseteq B, Q=B \subseteq D, R=A \subseteq C$ contrapositive of $(P \wedge Q) \rightarrow R$ is $\sim R \rightarrow \sim(P \wedge Q)$
i.e. $\quad \sim R \rightarrow(\sim P \vee \sim Q)$

Hence, If $A \nsubseteq C$, then $A \nsubseteq B$ or $B \neq D$
Correct Option (4)

Example 5: The negation of $p \vee(\sim p \wedge q)$
1) $\sim p \wedge \sim q$
2) $p \wedge \sim q$
3) $\sim p \vee q$
4) $\sim p \vee \sim q$

Solution

$\\\begin{array}{c|cc} \mathbf{p} & \mathbf{q} & \neg(\mathbf{p} \vee(\neg \mathbf{p} \wedge \mathbf{q})) \\ \hline F & F & \mathbf{T} \\ \hline F & T & F \\ \hline T & F & F \\ \hline T & T & F \end{array}$

$\begin{array}{c|cc} \mathbf{p} & \mathbf{q} & (\neg \mathbf{p} \mathbf{\Lambda} \neg \mathbf{q}) \\ \hline \mathrm{F} & \mathrm{F} & \mathbf{T} \\ \hline \mathrm{F} & \mathrm{T} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{F} & \mathrm{F} \\ \hline \mathrm{T} & \mathrm{T} & \mathrm{F} \end{array}$

Correct Answer: Option A

Summary

The concepts of converse, inverse, and contrapositive are fundamental in understanding logical implications and reasoning. They allow us to explore the relationships between propositions systematically and form the basis of many logical proofs and arguments. While the original statement and its contrapositive are always equivalent, the converse and inverse require careful consideration as they may not share the same truth value as the original statement.