What is the difference between the inverse and the contrapositive?

Both negate the two parts, but the inverse keeps the order ($\neg p \to \neg q$) while the contrapositive also swaps it ($\neg q \to \neg p$). The contrapositive is equivalent to the original; the inverse is equivalent to the converse.

If the original statement is false, is the contrapositive false too?

Yes. They are equivalent, so they always share their truth value. "If a number is even, then it is divisible by $4$" is false ($6$ is a counterexample), and so is "If a number is not divisible by $4$, then it is not even", with the same counterexample $6$.

Is a contrapositive the same as a proof by contradiction?

No. Using the contrapositive means you prove $\neg q \to \neg p$ directly instead of $p \to q$. A proof by contradiction assumes the whole statement is false and reaches an impossible result. Both methods appear in the proofs topic.

Teacher Resources

Contrapositive

Q: Why is the contrapositive always true when the original is true?

The original $p \to q$ says: whenever $p$ happens, $q$ happens. So if $q$ did not happen, $p$ cannot have happened either, otherwise $q$ would have followed. That is exactly $\neg q \to \neg p$. The truth table confirms it in all four cases.

Learn how to form the contrapositive of an if-then statement, why it always has the same truth value as the original, and how to use it to reason about numbers.

Intermediate25 minLesson

Printable worksheet

Definition

The contrapositive of an if-then statement "If

p

, then

q

" (

p \to q

) is the statement "If not

q

, then not

p

\neg q \to \neg p

To build it you do two things: swap the hypothesis and the conclusion, and negate both of them.

Key fact: a statement and its contrapositive are logically equivalent. They are always both true or both false. The truth table shows it, row by row:

\begin{array}{c|c|c|c|c|c} p & q & \neg p & \neg q & p \to q & \neg q \to \neg p \\ \hline T & T & F & F & T & T \\ T & F & F & T & F & F \\ F & T & T & F & T & T \\ F & F & T & T & T & T \end{array}

The last two columns are identical, so we write

p \to q \equiv \neg q \to \neg p

. (The interactive table in this lesson checks this for you: its last column,

(p \to q) \leftrightarrow (\neg q \to \neg p)

, is T in every row.)

The four related statements of

p \to q

Original: $p \to q$
Converse: $q \to p$ (swap only)
Inverse: $\neg p \to \neg q$ (negate only)
Contrapositive: $\neg q \to \neg p$ (swap and negate)

The original and the contrapositive always agree. The converse and the inverse also always agree with each other, because the inverse is the contrapositive of the converse.

Try it now

What is the contrapositive of

p \to q

Worked Examples

Write the contrapositive of: "If a number is divisible by $10$ , then it ends in $0$ ."

Identify the hypothesis $p$ and the conclusion $q$

$p$ : a number is divisible by $10$ . $q$ : it ends in $0$ . → Two parts found

Negate both parts

$\neg p$ : the number is not divisible by $10$ . $\neg q$ : it does not end in $0$ . → Both parts negated

Swap the order and write $\neg q \to \neg p$

Put the negated conclusion first and the negated hypothesis second. → If a number does not end in $0$ , then it is not divisible by $10$ .

Answer: If a number does not end in $0$ , then it is not divisible by $10$ . Like the original, this statement is true: for example, $73$ does not end in $0$ and $73 \div 10 = 7.3$ is not a whole number.

Common Mistakes

Only swapping the parts and calling $q \to p$ the contrapositive

Why it's wrong: Swapping without negating gives the converse, which can be false even when the original is true.

Correct: Swap and negate both parts: the contrapositive of $p \to q$ is $\neg q \to \neg p$ .

Negating " $x > 5$ " as " $x < 5$ "

Why it's wrong: This forgets the case $x = 5$ . Every number is either greater than $5$ or not greater than $5$ , and "not greater" includes $5$ itself.

Correct: The negation of $x > 5$ is $x \le 5$ , and the negation of $x \ge 5$ is $x < 5$ .

Assuming the converse of a true statement is also true

Why it's wrong: "If $x = 4$ , then $x^2 = 16$ " is true, but "If $x^2 = 16$ , then $x = 4$ " is false, because $x = -4$ also works.

Correct: Only the contrapositive is guaranteed to share the truth value of the original. The converse must be checked separately.

Getting lost in double negatives

Why it's wrong: For "If it is not raining, then I walk to school", students write "If I do not walk, then it is not not raining" or even drop a negation by accident.

Correct: Negating "not raining" gives "raining". The contrapositive is "If I do not walk to school, then it is raining."

Interactive Visual

Truth Table

Expressionp → q ↔ ¬q → ¬p

p	q	p → q	¬q	¬p	¬q → ¬p	p → q ↔ ¬q → ¬p
T	T	T	F	F	T	T
T	F	F	T	F	F	T
F	T	T	F	T	T	T
F	F	T	T	T	T	T

T= TrueF= False

Truth Table

Select Operator

p → q

p	q	p → q
T	T	T
T	F	F
F	T	T
F	F	T

T= TrueF= False

Interactive Sandbox

Expression Calculator

Try these:

History

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Practice Problems

18 problems

Problem 1 of 18

Easy

What is the contrapositive of $p \to q$ ?

Practice Problems: Contrapositive

1. What is the contrapositive of p → q?
2. Rule: if a number is a multiple of 4, then it is even. Its contrapositive says: if a number is not even, then it is not a multiple of 4. Which of the numbers 12, 15, 20, 28 can you rule out with the contrapositive alone?
3. Rule: if a number is divisible by 10, then its last digit is 0. Use the contrapositive to decide whether 7345 is divisible by 10.
4. Which statement is the contrapositive of "If a number ends in 5, then it is divisible by 5"?
5. A conditional statement is true. Besides the statement itself, how many of the other three related statements (converse, inverse, contrapositive) are guaranteed to be true as well?
6. Statement: if x = 3, then 2x = 6. Its contrapositive is: if 2x ≠ 6, then x ≠ 3. You know that 2x = 14. Show that x is not 3 and find x.
7. A conditional statement p → q is known to be true. Which related statement must also be true?
8. Use 1 for true and 0 for false. In the truth table row where p = 1 and q = 0, find the values of the contrapositive and of the original.
9. A truth table for \neg q \to \neg p has 4 rows. In how many of these rows is \neg q \to \neg p true?
10. Use 1 for true and 0 for false. In the row where p = 0 and q = 1, the original p → q is 1. Find the contrapositive and the converse in this row.
11. In the truth table row where p is true and q is false, what are the values of p \to q and \neg q \to \neg p?
12. Rule: if a number is a multiple of 6, then it is a multiple of 3. To show with the contrapositive that 50 is not a multiple of 6, you check that 50 is not a multiple of 3. What is the remainder when 50 is divided by 3?
13. Rule: if a number is divisible by 3, then its digit sum is divisible by 3. Use the contrapositive to decide whether 2471 is divisible by 3.
14. Original: "If x = 4, then x² = 16." Which of the four related statements (original, converse, inverse, contrapositive) are true?
15. The statement "If 3x + 2 > 20, then x > 6" is true. Its contrapositive is "If x ≤ 6, then 3x + 2 ≤ 20." Check the contrapositive for x = 6 and x = 5.
16. The contrapositive of "If n² is even, then n is even" is "If n is odd, then n² is odd." Check it for n = 13: write 13² in the form 2m + 1. What is m?
17. To see why "If n² is even, then n is even" holds, check its contrapositive "If n is odd, then n² is odd" for n = 9, using the form n = 2k + 1.
18. For whole numbers a, b ≥ 0: "If a + b ≥ 10, then a ≥ 5 or b ≥ 5." Its contrapositive is "If a < 5 and b < 5, then a + b < 10." Find the largest possible value of a + b when a < 5 and b < 5.

Why It Matters

The contrapositive gives you a second door into the same statement, and sometimes that door is much easier to open:

Number facts: To show that $2750$ is not divisible by $6$ , it is enough to see that it is not divisible by $3$
Checking work: If a correct answer must be a whole number and you get $4.5$ , you know something went wrong
Computer programs: "If the password is correct, the door opens." The door stayed closed, so the password was wrong
Proofs: Later you will prove statements like "If $n^2$ is even, then $n$ is even" by proving the contrapositive instead

Knowing which related statements agree also protects you from a very common error: assuming that the converse of a true statement must be true.

Real World Applications

Divisibility Checks and Check Digits

Many codes and quick mental checks rely on divisibility rules, and the contrapositive is what lets you reject a number without dividing.

Example:

Rule: if a number is divisible by $9$ , its digit sum is divisible by $9$ . The digit sum of $5812$ is $16$ , so $5812$ cannot be divisible by $9$ .

1Try It Yourself

Rule: if a number is divisible by $9$ , then its digit sum is divisible by $9$ . A clerk checks the code number $4372$ .

Is $4372$ divisible by $9$ ?

Step 1: Write the mathematical expression

Write the digit sum of $4372$ as an addition:

Free Shipping Rules

Online shops often have rules like "If your order is 50 dollars or more, shipping is free." The contrapositive tells you what a shipping fee on your receipt means.

Example:

Contrapositive: if you paid for shipping, your order was less than 50 dollars. A receipt with a shipping fee proves the order was under the limit.

2Try It Yourself

Rule: if an order totals 50 dollars or more, shipping is free. Anna paid a shipping fee. Her basket had a book for 18 dollars and a lamp for 27 dollars.

What was her order total, and how much more would she have needed for free shipping?

Step 1: Write the mathematical expression

Write the order total as an addition:

Reading League Tables

Sports fans use the contrapositive to rule things out from the points table without watching every game.

Example:

In a league with $3$ points for a win, $1$ for a draw and $0$ for a loss: if a team won all $5$ games, it has $15$ points. A team with fewer than $15$ points did not win every game.

3Try It Yourself

Rule: if a team wins all $5$ of its games, then it has $15$ points ( $3$ per win, $1$ per draw, $0$ per loss). Team Blue has $13$ points after $5$ games.

Did Team Blue win all its games, and at most how many games could it have won?

Step 1: Write the mathematical expression

Write the points for $5$ wins as a multiplication:

Key Takeaways

1The contrapositive of $p \to q$ is $\neg q \to \neg p$ : swap the two parts and negate both
2A statement and its contrapositive are logically equivalent: they are always both true or both false
3The truth table shows it: $p \to q$ and $\neg q \to \neg p$ are both false only in the row $p$ true, $q$ false
4The converse $q \to p$ and the inverse $\neg p \to \neg q$ agree with each other, but not necessarily with the original
5Negate carefully: the opposite of $x > 5$ is $x \le 5$ , and the opposite of odd is even
6When a statement is hard to check directly, check its contrapositive instead, for example "if $n$ is odd, then $n^2$ is odd"

Frequently Asked Questions

The original

p \to q

says: whenever

p

happens,

q

happens. So if

q

did not happen,

p

cannot have happened either, otherwise

q

would have followed. That is exactly

\neg q \to \neg p

. The truth table confirms it in all four cases.

Glossary

Contrapositive: The statement $\neg q \to \neg p$ formed from $p \to q$ by swapping and negating both parts
Converse: The statement $q \to p$ , formed by swapping hypothesis and conclusion
Inverse: The statement $\neg p \to \neg q$ , formed by negating hypothesis and conclusion without swapping
Negation: The opposite of a statement, written $\neg p$ ; it is true exactly when $p$ is false
Logically equivalent: Two statements that have the same truth value in every row of a truth table, written with $\equiv$

Formula Card

Contrapositive

p \to q \equiv \neg q \to \neg p

Swap and negate both parts; always equivalent to the original

Converse and inverse

q \to p \equiv \neg p \to \neg q

Equivalent to each other, but not necessarily to the original

Negating inequalities

\neg(x > a) \iff x \le a

The negation of "greater than" is "less than or equal to"

Contrapositive

Definition

Worked Examples

Common Mistakes

Interactive Visual

Truth Table

Truth Table

Interactive Sandbox

Expression Calculator

History

Practice Problems

Why It Matters

Real World Applications

Divisibility Checks and Check Digits

Free Shipping Rules

Reading League Tables

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Contrapositive

Converse and inverse

Negating inequalities

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Converse and Inverse