Teacher Guide: Contrapositive

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.

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For Teachers

Learning Objectives

Form the contrapositive of a conditional statement by swapping and negating both parts
Use a truth table to show that a statement and its contrapositive are logically equivalent
Write the original, converse, inverse and contrapositive side by side and identify which pairs agree
Negate statements correctly, including inequalities and double negatives
Use the contrapositive to reason about numbers, such as divisibility and parity

Prerequisites

• If-then statements: hypothesis, conclusion and when an implication is true (If-Then Statements lesson)
• Converse and inverse of a conditional statement (Converse and Inverse lesson)
• Truth tables and the negation NOT (Truth Tables and The NOT Operation lessons)
• Divisibility rules for 3, 9 and 10, and even and odd numbers

Discussion Starters

1. "If you live in Paris, then you live in France." What can you conclude about someone who does not live in France?
2. Why does swapping AND negating keep the meaning of a statement, while doing only one of them can change it?
3. Can you find a true statement whose converse is also true? Does that break the rule that the converse is not equivalent?
4. Why might a mathematician choose to work with the contrapositive instead of the original statement?

Common Misconceptions

The contrapositive and the inverse are the same because both use "not"

Remediation: Put both next to each other with the same example (squares and rectangles) and let students find a counterexample to the inverse. Stress that the order of the parts matters.

A statement that is true for many examples has a true converse

Remediation: Use number examples like "If $x = 4$ , then $x^2 = 16$ " and ask students to search for a counterexample to the converse. One counterexample is enough to make it false.

The negation of "greater than" is "less than"

Remediation: Draw a number line and shade $x > 5$ . Ask what is left unshaded: everything up to and including $5$ , so the negation is $x \le 5$ .

Differentiation Ideas

For Struggling Students:

• Start with everyday statements (rain and wet streets) before using symbols
• Give a template: If not ___, then not ___, and have students fill in the conclusion first
• Color code the hypothesis and the conclusion so the swap is visible

For On-Level Students:

• Write all four related statements for divisibility rules and decide which are true
• Complete truth tables for p implies q, its converse, inverse and contrapositive
• Use contrapositives to rule out numbers in divisibility puzzles

For Advanced Students:

• Explain why if n squared is even then n is even, using the contrapositive
• Form contrapositives of statements with and and or, using De Morgan's laws
• Compare using the contrapositive with proof by contradiction on the same statement

Standards Alignment

MP3 (CCSS.MATH.PRACTICE.MP3)
Construct viable arguments and critique the reasoning of others
MP6 (CCSS.MATH.PRACTICE.MP6)
Attend to precision
HSG.CO.C.9 (CCSS.MATH.CONTENT.HSG.CO.C.9)
Prove theorems about lines and angles (these proofs are built from conditional statements)

Lesson Resources

visualInteractive Truth Table
Students explore the NOT and IMPLIES tables, then use them to build the column for not q implies not p and compare it with p implies q
activityFour Statements Card Sort
Groups sort cards into original, converse, inverse and contrapositive and mark each true or false
worksheetContrapositive Number Detective
Use contrapositives of divisibility rules to rule out numbers without dividing

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

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.

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."

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

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.

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.

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"

Teacher Guide: Contrapositive

Use this lesson with your class

For Teachers

For Struggling Students:

For On-Level Students:

For Advanced Students:

Lesson Content

Definition

Worked Examples

Common Mistakes

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