Converse and Inverse

Learn how to form the converse and the inverse of an if-then statement and why a true statement can have a false converse.

Intermediate25 min18 problems
Practice Problems

Definition

Every conditional statement has the form p→qp \to q: if pp (the hypothesis), then qq (the conclusion). From it we can build two new statements:
  • The converse swaps the two parts: q→pq \to p (if qq, then pp).
  • The inverse negates both parts: ¬p→¬q\neg p \to \neg q (if not pp, then not qq).
Example: Original: If a number is divisible by 4, then it is even.
  • Converse: If a number is even, then it is divisible by 4. (False: 6 is even, but 6÷4=1.56 \div 4 = 1.5.)
  • Inverse: If a number is not divisible by 4, then it is not even. (False: 6 again.)
The truth table shows that the converse and the inverse do not always match the original, but they always match each other:
pqp→qq→p¬p→¬qTTTTTTFFTTFTTFFFFTTT\begin{array}{cc|c|c|c} p & q & p \to q & q \to p & \neg p \to \neg q \\ \hline \text{T} & \text{T} & \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} & \text{T} & \text{T} \\ \text{F} & \text{T} & \text{T} & \text{F} & \text{F} \\ \text{F} & \text{F} & \text{T} & \text{T} & \text{T} \end{array}
The rows can be listed in any order: the interactive table below starts with the row where pp and qq are both false, and each row keeps the same values.
To show that a statement is false, one counterexample is enough. When a statement and its converse are both true, we combine them into a biconditional: p↔qp \leftrightarrow q, read as pp if and only if qq.

Try it now

What is the converse of the conditional p→qp \to q?

Worked Examples

Original: If a shape is a square, then it has four sides. Write the converse and decide whether it is true.
1

Identify the hypothesis and the conclusion

pp: a shape is a square. qq: it has four sides. → Original: p→qp \to q (true)
2

Swap the two parts

q→pq \to p: If a shape has four sides, then it is a square. → This is the converse
3

Look for a counterexample

A rectangle that is 5 cm by 3 cm has four sides but is not a square. → The converse is false

Common Mistakes

Assuming the converse is true because the original is true

Why it's wrong: Swapping hypothesis and conclusion creates a new statement with its own truth value. If x=3x = 3, then x2=9x^2 = 9 is true, but if x2=9x^2 = 9, then x=3x = 3 is false.

Correct: Always test the converse on its own and look for a counterexample, here x=−3x = -3.

Mixing up the converse and the inverse

Why it's wrong: Both are built from the same two parts, so they look alike. The converse changes the order, the inverse adds two negations.

Correct: Converse = swap: q→pq \to p. Inverse = negate both: ¬p→¬q\neg p \to \neg q.

Negating only one part when forming the inverse

Why it's wrong: Writing ¬p→q\neg p \to q (if not a square, then four sides) is not the inverse. It is a completely different statement.

Correct: Put not in front of the hypothesis and the conclusion: if not a square, then not four sides.

Using one example that works as proof that a converse is true

Why it's wrong: For if x2=9x^2 = 9, then x=3x = 3, the example x=3x = 3 works, but x=−3x = -3 does not. Examples that work never prove a general statement.

Correct: One counterexample proves a statement false. To prove it true you need an argument that covers every case.

Interactive Visual

Truth Table

p → q
pqp → q
FFT
FTT
TFF
TTT

T= TrueF= False

Truth Table

p → q
pqp → q
FF
FT
TF
TT

T= TrueF= False

Interactive Sandbox

Expression Calculator

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Practice Problems

18 problems
Problem 1 of 18
Easy
What is the converse of the conditional p→qp \to q?

Why It Matters

Mixing up a statement and its converse is one of the most common errors in reasoning:
  • Mathematics: A theorem being true does not make its converse true. The Pythagorean theorem and its converse are proven separately.
  • Definitions: A good definition is a biconditional. It must work in both directions.
  • Advertising: If you buy the premium plan, you get faster service. That does not mean everyone with fast service bought the premium plan.
  • Programming: If x is divisible by 6, then x is even. A program must not assume the reverse: 8 is even but not divisible by 6.
  • Everyday arguments: If it rains, the street is wet. A wet street does not prove it rained, a street cleaner may have passed.
Spotting a false converse with one counterexample is a skill you will use in every proof.

Real World Applications

Shop Promotions

Shops state rules as if-then statements. Customers often read them backwards, which is exactly the converse.

Example:

Rule: If your order is over 50 dollars, then delivery is free. Converse: If delivery is free, then your order was over 50 dollars. A member code can also give free delivery, so the converse can be false.

1Try It Yourself

A shop rule says: If an order is over 50 dollars, then delivery is free. Lena buys 3 books at 12 dollars each and uses a member code that also gives free delivery.

Does Lena's order show that the converse of the rule is false?

Step 1: Write the mathematical expression

Write the expression for Lena's total:

Checking Rules in Computer Programs

Programmers write conditions like if n is divisible by 6, then n is even. Reversing such a condition is a classic bug.

Example:

A program accepts only numbers divisible by 6 and labels them even. A programmer flips the check and accepts every even number as divisible by 6. The number 8 slips through by mistake.

2Try It Yourself

The rule is: If a number is divisible by 6, then it is even. A programmer uses the converse: If a number is even, then it is divisible by 6. Test it with the number 8.

Does 8 break the converse?

Step 1: Write the mathematical expression

Write the division that tests 8 against 6:

Good Definitions

Dictionaries and math books write definitions that work in both directions. A definition is a biconditional, an if and only if statement.

Example:

A triangle is a right triangle if and only if it has a 90∘90^\circ angle. Every right triangle has a 90∘90^\circ angle, and every triangle with a 90∘90^\circ angle is a right triangle.

3Try It Yourself

Two angles of a triangle measure 35∘35^\circ and 55∘55^\circ. The angles of a triangle add up to 180∘180^\circ.

Is it a right triangle?

Step 1: Write the mathematical expression

Write the expression for the third angle:

Key Takeaways

  • 1A conditional statement has the form p→qp \to q: if pp, then qq
  • 2The converse swaps the parts: q→pq \to p
  • 3The inverse negates both parts: ¬p→¬q\neg p \to \neg q
  • 4A true statement can have a false converse and a false inverse, for example if x=3x = 3, then x2=9x^2 = 9
  • 5The converse and the inverse always have the same truth value, and one counterexample is enough to show a statement is false
  • 6When a statement and its converse are both true, they form a biconditional p↔qp \leftrightarrow q (if and only if)

Frequently Asked Questions

Sometimes. If an angle measures 90∘90^\circ, then it is a right angle, and the converse is also true. But you always have to check it separately, it never follows automatically.
Sometimes. If an angle measures 90∘90^\circ, then it is a right angle, and the converse is also true. But you always have to check it separately, it never follows automatically.
Look at the truth table: the columns for q→pq \to p and ¬p→¬q\neg p \to \neg q are identical in all four rows. In the next lesson you will see why: each one is the contrapositive of the other.
It means both directions are true: p→qp \to q and q→pq \to p. We write it p↔qp \leftrightarrow q. For example, x+4=10x + 4 = 10 if and only if x=6x = 6.
Just one. A single case where the hypothesis is true and the conclusion is false shows the whole statement is false.

Glossary

Conditional statement
A statement of the form if pp, then qq, written p→qp \to q
Converse
The statement formed by swapping the hypothesis and the conclusion: q→pq \to p
Inverse
The statement formed by negating both the hypothesis and the conclusion: ¬p→¬q\neg p \to \neg q
Negation
The opposite of a statement, written ¬p\neg p and read not pp
Counterexample
A single case where the hypothesis is true but the conclusion is false
Biconditional
A statement that is true in both directions, written p↔qp \leftrightarrow q and read pp if and only if qq

Formula Card

Original (conditional)

p→qp \to q

If $p$, then $q$

Converse

q→pq \to p

Swap the hypothesis and the conclusion

Inverse

¬p→¬q\neg p \to \neg q

Negate both parts, keep the order

Biconditional

p↔qp \leftrightarrow q

True when $p \to q$ and $q \to p$ are both true

Today's challengeOne problem, three tries

More in This Topic

Introduction to Logical Reasoning

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Deductive Reasoning

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If-Then Statements

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Contrapositive

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