Converse
The implication is not a commutative operation. We get the converse of an implication by swapping its operands; that is, by swapping the roles of the hypothesis and conclusion.
Take $p$ and $q$ as two arbitrary propositions and let $I$ denote the implication with premise $p$ and conclusion $q$; that is,
$I : p \Rightarrow q.$
Now swap the operands to make $C$ as:
$C : q \Rightarrow p.$
We define $C$ to be the converse of $I$.
$C = \mathrm{Converse}\left(I\right).$
If we look closely, we notice that $I$ is also converse of $C$. That means $I$ and $C$ are converses of each other.
Note: $I$ and $C$ are not logically equivalent.
Examples
Let’s look at some examples.
Consider the following propositions:
 $m_h$: Murphy is a pet horse.
 $m_l$: Murphy has four legs.
Now,
 $I_m \equiv m_h \Rightarrow m_l$
$I_m$ states that if Murphy is a pet horse, then Murphy has four legs.

$C_m \equiv \mathrm{Converse}\left( I_m\right)$ $\equiv m_{l}\Rightarrow m_{h}.$

$C_m$ states that if Murphy has four legs, then Murphy is a pet horse.
Both $I_m$ and $C_m$ are converse of each other. Furthermore, they are not logically equivalent.
To understand it further, let’s assume that Murphy is a pet dog. In that case, $I_m$ is true because hypothesis ($m_h$) is false. While $C_{m}$ is false because hypothesis ($m_{l}$) is true and the conclusion ($m_{h}$) is false.
Let’s take a look at another example.
Consider:
 $L_{C}$: Lee lives in China.
 $L_{A}$: Lee lives in Asia.
Now,

$I_{L} = L_{C} \Rightarrow L_{A}$: If Lee lives in China then Lee lives in Asia.

$C_{L} = \text{Converse} \left(I_{L}\right) = L_{A} \Rightarrow L_{C}.$

$C_{L}$: If Lee lives in Asia, then Lee lives in China.
Here, $I_{L}$ and $C_{L}$ are converse of each other.
To see the difference between $I_{L}$ and $C_{L}$ assume that Lee lives in Thailand. In that case, $I_L$ is true while $C_L$ is false.
Inverse
If we negate both the operands of an implication without changing the direction, we get the inverse of it.
Take $p$ and $q$ as two arbitrary propositions and make $q_1$ as:
$q_1 = p \Rightarrow q.$
Now negate both the operands to make $q_3$ as:
$q_3 = \neg p \Rightarrow \neg q.$
We define $q_3$ to be the inverse of $q_1$.
$q_3 = \text{Inverse}\left(q_1\right).$
If we look closely, we notice that $q_1$ is also the inverse of $q_3$. That is, $q_1$ and $q_3$ are inverse of each other. Further, a proposition and its inverse are not logically equivalent.
Note: $q_1$ and $q_3$ are not logically equivalent.
Examples
Let’s see what is the inverse of $C_{L}$.
Recall that,
 $C_{L} = L_{A} \Rightarrow L_{C}$: If Lee lives in Asia then Lee lives in China.
Now,
 $V_{C}=\text{Inverse}\left(C_{L}\right)$ $= \neg L_{A}\Rightarrow \neg L_{C}.$
 $V_{C}$: If Lee does not live in Asia, then Lee does not live in China.
Let us compare it with the inverse of $I_{L}$, which is:
 $I_{L} = L_{C} \Rightarrow L_{A} :$ If Lee lives in China, then Lee lives in Asia.
 $V_{I}= \text{Inverse}\left(I_{L}\right)$ $= \neg L_{C}\Rightarrow \neg L_{A}.$
 $V_{I}$: If Lee does not live in China, then Lee does not live in Asia.
Note that if Lee lives in Thailand, then $V_{I}$ is false while $V_C$ is true.
Let’s look at another simple example.
Consider the following propositions:
 $s_l$: Sana has taken a literature examination.
 $s_{m}$: Sana secured 75% marks in the literature examination.
Now,
 $I_{s} = s_{l} \Rightarrow s_{m}$: If Sana has taken a literature examination, then she has secured 75% marks in it.
 $V_{s} = \text{Inverse}\left(I_{s}\right)$ $= \neg s_{l} \Rightarrow \neg s_{m}.$
 $V_{s}$: If Sana has not taken a literature examination, then she has not secured 75% marks in it.
Note that if Sana had secured 75% marks in literature examination without taking a literature examination then $I_s$ is true and $V_s$ is false. Similarly, we can think of the scenario when Sana had taken literature examination and had not secured 75% marks. In that case, $I_s$ is false and $V_s$ is true.
Contrapositive
We can get contrapositive of an implication by changing its direction and negating both of its operands.
Take $p$ and $q$ as two arbitrary propositions and make $q_1$ as:
$q_1 = p \Rightarrow q.$
Now negate both the operands and swap them to make $q_4$ as:
$q_4 = \neg q \Rightarrow \neg p.$
We define $q_4$ to be the contrapositive of $q_1$.
$q_4 =\text{Contrapositive}\left(q_1\right).$
If we look closely, we notice that $q_1$ is also contrapositive of $q_4$. Hence, $q_1$ and $q_4$ are contrapositive of each other. Further, it is essential to note that $q_1$ and $q_4$ are logically equivalent.
Note: An implication and its contrapositive are logically equivalent.
Examples
Let’s look at some examples.
Recall that,
 $m_{h}$: Murphy is a pet horse.
 $m_{l}$: Murphy has four legs.
Now,
 $I_{m}= m_{h}\Rightarrow m_{l}$: If Murphy is a pet horse, then Murphy has four legs.
 $P_{m}= \text{Contrapositive}\left(I_{m}\right)$ $=\neg m_{l}\Rightarrow \neg m_{h}.$
 $P_{m}$: If Murphy does not have four legs, then Murphy is not a pet horse.
Both $I_{m}$ and $P_{m}$ are contrapositive of each other. Further, both are logically equivalent.
For the next example:
Recall that,
 $L_{C}$: Lee lives in China.
 $L_{A}$: Lee lives in Asia.
 $I_{L} = L_{C} \Rightarrow L_{A}$: If Lee lives in China, then Lee lives in Asia.
Let’s look at the contrapositive of $I_{L}$.
 $P_{L} =\text{Contrapositive}\left(I_{L}\right)$ $= \neg L_{A} \Rightarrow \neg L_{C}.$
 $P_{L}$: If Lee does not live in Asia, then Lee does not live in China.
Here, $I_{L}$ and $P_{L}$ are contrapositive of each other, and they are logically equivalent.
Let’s take another example.
Consider the following propositions:
 $ET$: Emma is taller than Charlotte.
 $CS$: Charlotte is shorter than Emma.
Now,

$I_{EC} = ET \Rightarrow CS$: If Emma is taller than Charlotte, then Charlotte is shorter than Emma.

$P_{EC} =\text{Contrapositive} \left(I_{EC}\right)$ $= \neg CS\Rightarrow \neg ET.$

$P_{EC}$: If Charlotte is not shorter than Emma, then Emma is not taller than Charlotte.
Summary
Take $p$ and $q$ as two arbitrary propositions and make $q_1$ as:
$q_1 = p \Rightarrow q.$

$\text{Converse}\left(q_1\right):= q \Rightarrow p.$

$\text{Inverse}\left(q_1\right):= \neg p \Rightarrow \neg q.$

$\text{Contrapositive}\left(q_1\right):= \neg q \Rightarrow \neg p.$
Note: The $:=$ symbol used above means “defined as” or “is by definition equal to.”
Quiz
Test your understanding of the implication.
Select the contrapositive of the proposition $q_1 \Rightarrow q_2$ from the given options.
$q_2 \Rightarrow q_1$
$\neg q_1 \Rightarrow \neg q_2$
$\neg q_2 \Rightarrow \neg q_1$
$q_1 \Rightarrow \neg q_2$