Derived Implication

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.

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.

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 $C_{L}$.

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Recall that,

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

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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.

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.

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Recall that,

• $m_{h}$: Murphy is a pet horse.
• $m_{l}$: Murphy has four legs.

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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:

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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.

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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.

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