Converse, Inverse, and Contrapositive Truth Table

Q: What is converse of a statement?

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

Q: What is inverse of a statement?

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.”

Q: What is contrapositive of a statement?

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.”

Converse, Inverse, and Contrapositive

Edited By Komal Miglani | Updated on Jul 02, 2025 07:55 PM IST

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

Statement If p, then qp→q Converse If q, then pq→p Inverse If not p, then not q(∼p)→(∼q) Contrapositive If not q, then not p(∼q)→(∼p)

Note:

A given statement and its contrapositive have the same meaning
As the inverse is the contrapositive of the converse, so it has the same meaning as the converse
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.”

Statement If p, then qp→q Converse If q, then pq→p Inverse If not p, then not q(∼p)→(∼q) Contrapositive If not q, then not p(∼q)→(∼p)

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.”

Statement If p, then qp→q Converse If q, then pq→p Inverse If not p, then not q(∼p)→(∼q) Contrapositive If not q, then not p(∼q)→(∼p)

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.”

Statement If p, then qp→q Converse If q, then pq→p Inverse If not p, then not q(∼p)→(∼q) Contrapositive If not q, then not p(∼q)→(∼p)

If the statement is p→q contrapositive is ∼q→∼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⊆B and B⊆D, then A⊆C′′ is :
1) If A⊆C, then B⊂A or D⊂B
2) If A⊈C, then A⊆B and B⊆D
3) If A⊈C, then A⊈B and B⊆D
4) If A⊈C, then A⊈B or B⊈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.”

Statement If p, then qp→q Converse If q, then pq→p Inverse If not p, then not q(∼p)→(∼q) Contrapositive If not q, then not p(∼q)→(∼p)

Let P=A⊆B,Q=B⊆D,R=A⊆C contrapositive of (P∧Q)→R is ∼R→∼(P∧Q)
i.e. ∼R→(∼P∨∼Q)

Hence, If A⊈C, then A⊈B or B≠D
Correct Option (4)

Example 5: The negation of p∨(∼p∧q)
1) ∼p∧∼q
2) p∧∼q
3) ∼p∨q
4) ∼p∨∼q

Solution

pq¬(p∨(¬p∧q))FFTFTFTFFTTF

pq(¬pΛ¬q)FFTFTFTFFTTF

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.

Frequently Asked Questions (FAQs)

1. What is a statement?

A sentence is called a mathematical statement if it is either true or false but not both.

2. What is a conditional statement?

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”

3. What is converse of a statement?

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

4. What is inverse of a statement?

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.”

5. What is contrapositive of a statement?

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.”

6. What is the converse of a statement?

The converse of a statement is formed by switching the hypothesis and conclusion of the original statement. For example, if the original statement is "If A, then B," the converse would be "If B, then A." It's important to note that the converse is not always true even if the original statement is true.

7. How does the inverse of a statement differ from the converse?

The inverse of a statement is formed by negating both the hypothesis and conclusion of the original statement. If the original statement is "If A, then B," the inverse would be "If not A, then not B." Unlike the converse, which switches the parts, the inverse keeps the order but negates both parts.

8. What is the contrapositive of a statement?

The contrapositive of a statement is formed by negating and switching the hypothesis and conclusion of the original statement. For a statement "If A, then B," the contrapositive would be "If not B, then not A." The contrapositive is logically equivalent to the original statement, meaning they are always either both true or both false.

9. What is the relationship between a statement's inverse and contrapositive?

The inverse and contrapositive of a statement are actually converses of each other. If we start with "If A, then B," the inverse is "If not A, then not B," and the contrapositive is "If not B, then not A." You can see that the inverse and contrapositive have their hypothesis and conclusion switched, making them converses of each other.

10. Why is it important to understand the differences between converse, inverse, and contrapositive in mathematical reasoning?

Understanding these concepts is crucial in mathematical reasoning because they help in constructing valid arguments and proofs. Knowing that the contrapositive is logically equivalent to the original statement allows for alternative proof strategies. Recognizing that the converse and inverse are not necessarily true prevents logical fallacies in arguments.

11. Why is the contrapositive always logically equivalent to the original statement?

The contrapositive is always logically equivalent to the original statement because it maintains the same logical relationship between the parts of the statement. By negating and switching the hypothesis and conclusion, the contrapositive preserves the original implication in a different form. This equivalence is a fundamental principle in logic and is used in various mathematical proofs.

12. Can the converse of a true statement be false?

Yes, the converse of a true statement can be false. The truth of the original statement does not guarantee the truth of its converse. For example, the statement "If it's raining, then the ground is wet" is true, but its converse "If the ground is wet, then it's raining" can be false (e.g., the ground could be wet due to a sprinkler).

13. How can you determine if a statement and its converse are both true?

To determine if a statement and its converse are both true, you need to evaluate each independently. If both the original statement "If A, then B" and its converse "If B, then A" are true, then A and B are equivalent conditions. This situation is relatively rare and is called a biconditional statement, often expressed as "A if and only if B."

14. How can you use the contrapositive in proving statements?

The contrapositive can be used as an alternative method for proving statements. Since the contrapositive is logically equivalent to the original statement, proving the contrapositive is the same as proving the original statement. This is particularly useful when the contrapositive is easier to prove or provides a more straightforward logical path.

15. What is a common mistake students make when dealing with converse statements?

A common mistake is assuming that if a statement is true, its converse must also be true. This is not always the case. Students often confuse the logical relationship between a statement and its converse, leading to incorrect conclusions. It's important to evaluate the converse independently from the original statement.

16. How do converse, inverse, and contrapositive relate to necessary and sufficient conditions?

These concepts help clarify the relationship between necessary and sufficient conditions:

17. Why is it important to consider the context when interpreting converse, inverse, and contrapositive in real-world situations?

Considering context is crucial because:

18. How does understanding these concepts contribute to developing critical thinking skills?

Understanding converse, inverse, and contrapositive enhances critical thinking by:

19. How do converse, inverse, and contrapositive relate to the concept of logical equivalence?

Regarding logical equivalence:

20. Why is it important to distinguish between correlation and causation when dealing with converse statements?

Distinguishing between correlation and causation is crucial because:

21. How can students use Venn diagrams or other visual aids to better understand the relationships between these logical concepts?

Students can use visual aids like:

22. How does negation work in forming the inverse and contrapositive?

Negation is key in forming both the inverse and contrapositive. For the inverse, you negate both the hypothesis and conclusion of the original statement. For the contrapositive, you negate and switch them. Proper negation often involves more than just adding "not" to a statement; it means expressing the opposite condition.

23. Can you provide an example of a statement where the original, converse, inverse, and contrapositive are all true?

Yes, here's an example: "A triangle is equilateral if and only if all its angles are 60 degrees." In this case:

24. How do converse, inverse, and contrapositive relate to logical implication?

These concepts are closely related to logical implication. The original statement represents a direct implication (If A, then B). The contrapositive (If not B, then not A) is an equivalent implication. The converse and inverse, however, are not equivalent implications and may or may not be true when the original statement is true.

25. Why is it incorrect to assume that disproving the converse disproves the original statement?

Disproving the converse does not disprove the original statement because they are logically independent. The original statement "If A, then B" does not claim that B only occurs when A is true. It only states that B follows from A, not that A is the only cause of B. Therefore, finding a case where B is true but A is false (disproving the converse) doesn't affect the truth of the original statement.

26. How can Venn diagrams help in understanding the relationships between a statement and its variations?

Venn diagrams can visually represent the logical relationships between sets described in statements. For a statement "If A, then B," A would be a subset of B in the diagram. This helps illustrate why the contrapositive is equivalent (the complement of B is a subset of the complement of A) and why the converse and inverse aren't necessarily true (B might extend beyond A).

27. What role do converse, inverse, and contrapositive play in formal logic and mathematical proofs?

These concepts are fundamental in formal logic and mathematical proofs. They provide different ways to express and analyze logical relationships. The contrapositive is particularly useful in proof by contradiction. Understanding these variations helps in constructing valid arguments, identifying logical fallacies, and developing alternative proof strategies.

28. How can you test if a real-world statement is logically equivalent to its contrapositive?

To test if a real-world statement is logically equivalent to its contrapositive, you can:

29. Why is it important to be cautious about using the inverse in arguments?

It's important to be cautious about using the inverse in arguments because the inverse is not logically equivalent to the original statement. Assuming the truth of the inverse based on the truth of the original statement is a logical fallacy known as the fallacy of the inverse. This can lead to incorrect conclusions and weak arguments in both mathematical and everyday reasoning.

30. Can you explain why the contrapositive is useful in proving statements by contradiction?

The contrapositive is useful in proofs by contradiction because:

31. How do converse, inverse, and contrapositive apply to "if and only if" statements?

For "if and only if" (biconditional) statements:

32. What is the difference between implication and equivalence in relation to these concepts?

Implication (If A, then B) means A is sufficient for B, but not necessarily equivalent. Equivalence (A if and only if B) means A and B imply each other. In terms of converse, inverse, and contrapositive:

33. How can understanding these concepts help in identifying logical fallacies in everyday arguments?

Understanding converse, inverse, and contrapositive helps identify logical fallacies by:

34. How do converse, inverse, and contrapositive relate to the concept of logical negation?

Logical negation is fundamental to these concepts:

35. Can you explain how the contrapositive can be used to reframe difficult "if-then" statements?

The contrapositive can reframe difficult "if-then" statements by:

36. What are some common real-world examples where misunderstanding these concepts leads to incorrect conclusions?

Common examples include:

37. How can teachers use the concepts of converse, inverse, and contrapositive to design effective math problems?

Teachers can design effective math problems by:

38. How can the study of converse, inverse, and contrapositive improve students' ability to construct valid mathematical proofs?

Studying these concepts improves proof construction by: