**Logical Connectives-**

Before you go through this article, make sure that you have gone through the previous article on **Logical Connectives**.

We have discussed-

- Logical connectives are the operators used to combine one or more propositions.
- In propositional logic. there are 5 basic connectives-

**Also Read-****Propositions**

In this article, we will discuss Converse, Inverse and Contrapositive of a conditional statement.

**Converse, Inverse and Contrapositive-**

For a conditional statement p → q,

- The converse statement is q → p
- The inverse statement ∼p → ∼q
- The contrapositive statement is ∼q → ∼p

**Important Notes-**

**Note-01:**

- For conditional statements (p → q) only, the converse, inverse and contrapositive statements can be written.

**Note-02:**

Performing any two actions always result in the third one. |

For example-

- Inverse of converse is contrapositive.
- Inverse of contrapositive is converse.
- Converse of inverse is contrapositive.
- Converse of contrapositive is inverse.
- Contrapositive of inverse is converse.
- Contrapositive of converse is inverse.

**Note-03:**

For a conditional statement p → q,

- Its converse statement (q → p) and inverse statement (∼p → ∼q) are equivalent to each other.
- p → q and its contrapositive statement (∼q → ∼p) are equivalent to each other.

**Also Read-****Converting English Sentences To Propositional Logic**

**PRACTICE PROBLEMS BASED ON CONVERSE, INVERSE AND CONTRAPOSITIVE-**

**Problem-01:**

Write the converse, inverse and contrapositive of the following statements-

- If today is Sunday, then it is a holiday.
- If 5x – 1 = 9, then x = 2.
- If it rains, then I will stay at home.
- I will dance only if you sing.
- I will go if he stays.
- We leave whenever he comes.
- You will qualify GATE only if you work hard.
- If you are intelligent, then you will pass the exam.

**Solution-**

**Part-01:**

We have-

- The given sentence is- “If today is Sunday, then it is a holiday.”
- This sentence is of the form- “If p then q”.

So, the symbolic form is **p → q** where-

p : Today is Sunday

q : It is a holiday

**Converse Statement-** If it is a holiday, then today is Sunday.

**Inverse Statement- **If today is not Sunday, then it is not a holiday.

**Contrapositive Statement-** If it is not a holiday, then today is not Sunday.

**Part-02:**

We have-

- The given sentence is- “If 5x – 1 = 9, then x = 2.”
- This sentence is of the form- “If p then q”.

So, the symbolic form is **p → q** where-

p : 5x – 1 = 9

q : x = 2

**Converse Statement-** If x = 2, then 5x – 1 = 9.

**Inverse Statement-** If 5x – 1 ≠ 9, then x ≠ 2.

**Contrapositive Statement-** If x ≠ 2, then 5x – 1 ≠ 9.

**Part-03:**

We have-

- The given sentence is- “If it rains, then I will stay at home.”
- This sentence is of the form- “If p then q”.

So, the symbolic form is **p → q** where-

p : It rains

q : I will stay at home

**Converse Statement-** If I will stay at home, then it rains.

**Inverse Statement-** If it does not rain, then I will not stay at home.

**Contrapositive Statement-** If I will not stay at home, then it does not rain.

**Part-04:**

We have-

- The given sentence is- “I will dance only if you sing.”
- This sentence is of the form- “p only if q”.

So, the symbolic form is **p → q** where-

p : I will dance

q : You sing

**Converse Statement-** If you sing, then I will dance.

**Inverse Statement-** If I will not dance, then you do not sing.

**Contrapositive Statement-** If you do not sing, then I will not dance.

**Part-05:**

We have-

- The given sentence is- “I will go if he stays.”
- This sentence is of the form- “q if p”.

So, the symbolic form is **p → q** where-

p : He stays

q : I will go

**Converse Statement-** If I will go, then he stays.

**Inverse Statement-** If he does not stay, then I will not go.

**Contrapositive Statement-** If I will not go, then he does not stay.

**Part-06:**

We have-

- The given sentence is- “We leave whenever he comes.”
- We can replace “whenever” with “if”.
- Then, the sentence is- “We leave if he comes.”
- This sentence is of the form- “q if p”.

So, the symbolic form is **p → q** where-

p : He comes

q : We leave

**Converse Statement-** If we leave, then he comes.

**Inverse Statement-** If he does not come, then we do not leave.

**Contrapositive Statement-** If we do not leave, then he does not come.

**Part-07:**

We have-

- The given sentence is- “You will qualify GATE only if you work hard.”
- This sentence is of the form- “p only if q”.

So, the symbolic form is **p → q** where-

p : You will qualify GATE

q : You work hard

**Converse Statement-** If you work hard, then you will qualify GATE.

**Inverse Statement-** If you will not qualify GATE, then you do not work hard.

**Contrapositive Statement-** If you do not work hard, then you will not qualify GATE.

**Part-08:**

We have-

- The given sentence is- “If you are intelligent, then you will pass the exam.”
- This sentence is of the form- “If p then q”.

So, the symbolic form is **p → q** where-

p : You are intelligent

q : You will pass the exam

**Converse Statement-** If you will pass the exam, then you are intelligent.

**Inverse Statement-** If you are not intelligent, then you will not pass the exam.

**Contrapositive Statement-** If you will not pass the exam, then you are not intelligent.

**Problem-02:**

What is the converse of the statement- “I stay only if you go”?

- I stay if you go.
- If I stay, then you go.
- If you do not go, then I do not stay.
- If I do not stay, then you go.

**Solution-**

- Try solving this problem yourself.
- Solution is in the linked video lecture.
- Option (A) is correct.

To gain better understanding about converting Converse, Inverse and Contrapositive,

**Next Article-****Tautology, Contradiction and Contingency**

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