Teacher Guide: 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.

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10 questions on Logical Reasoning. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Form the converse and the inverse of a conditional statement
Determine the truth value of a converse or an inverse and justify it
Use a single counterexample to show that a statement is false
Explain with a truth table why the converse and the inverse always share a truth value
Write a biconditional when a statement and its converse are both true

Prerequisites

• If-Then Statements: hypothesis, conclusion and the truth table of $p \to q$
• Deductive Reasoning: drawing conclusions from premises
• Negation of a simple statement (not $p$ )
• Divisibility, squares of negative numbers and basic angle facts

Discussion Starters

1. If all dogs are animals, are all animals dogs? What is the difference between the two sentences?
2. Can you think of an advertisement that tricks people into believing the converse?
3. Why is one counterexample enough to show a statement is false, while a hundred examples cannot prove it true?
4. Why must a good definition work in both directions?

Common Misconceptions

The converse of a true statement is always true

Remediation: Start with everyday statements like if an animal is a dog, then it is a mammal. Students quickly find mammals that are not dogs, such as cats. Then move to math: if $x = 3$ , then $x^2 = 9$ .

The inverse means the same as the original

Remediation: Show the truth table: the original and the inverse differ in rows 2 and 3. Test the divisible by 4 example with the number 6.

Converse and inverse are the same thing

Remediation: Color code: write $p$ in blue and $q$ in red. The converse changes the order of the colors, the inverse keeps the order and adds not to both.

Differentiation Ideas

For Struggling Students:

• Use sentence frames: If ___, then ___. Converse: If ___, then ___.
• Start with everyday statements before moving to numbers
• Give the counterexample and ask students to explain why it works

For On-Level Students:

• Write the converse and the inverse of five math statements and find counterexamples
• Fill in the truth table for $q \to p$ and $\neg p \to \neg q$
• Decide which statements can be written as biconditionals

For Advanced Students:

• Investigate the converse of the Pythagorean theorem and why it needs its own proof
• Find statements where the original is false but the converse is true
• Preview the contrapositive and compare its truth table with the original

Standards Alignment

8.G.B.6 (CCSS.MATH.CONTENT.8.G.B.6)
Explain a proof of the Pythagorean Theorem and its converse
MP3 (CCSS.MATH.PRACTICE.MP3)
Construct viable arguments and critique the reasoning of others
MP6 (CCSS.MATH.PRACTICE.MP6)
Attend to precision

Lesson Resources

visualInteractive Truth Table
Students compare the columns for $p \to q$ , $q \to p$ and $\neg p \to \neg q$ row by row
activityConverse Hunt
Pairs swap statement cards, write the converse and race to find a counterexample
worksheetTrue Both Ways?
Sort math statements into biconditionals and statements with a false converse

Lesson Content

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

Definition

Every conditional statement has the form

p \to q

: if

p

(the hypothesis), then

q

(the conclusion). From it we can build two new statements:

The converse swaps the two parts: $q \to p$ (if $q$ , then $p$ ).
The inverse negates both parts: $\neg p \to \neg q$ (if not $p$ , then not $q$ ).

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 $6 \div 4 = 1.5$ .)
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:

\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

p

and

q

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 \leftrightarrow q

, read as

p

if and only if

q

Worked Examples

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

Identify the hypothesis and the conclusion

$p$ : a shape is a square. $q$ : it has four sides. → Original: $p \to q$ (true)

Swap the two parts

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

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

Answer: Converse: If a shape has four sides, then it is a square. It is false, because a 5 cm by 3 cm rectangle is a counterexample.

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 $x = 3$ , then $x^2 = 9$ is true, but if $x^2 = 9$ , then $x = 3$ is false.

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

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: $q \to p$ . Inverse = negate both: $\neg p \to \neg q$ .

Negating only one part when forming the inverse

Why it's wrong: Writing $\neg p \to q$ (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 $x^2 = 9$ , then $x = 3$ , the example $x = 3$ works, but $x = -3$ 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.

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 $90^\circ$ angle. Every right triangle has a $90^\circ$ angle, and every triangle with a $90^\circ$ angle is a right triangle.

3Try It Yourself

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

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 $p \to q$ : if $p$ , then $q$
2The converse swaps the parts: $q \to p$
3The inverse negates both parts: $\neg p \to \neg q$
4A true statement can have a false converse and a false inverse, for example if $x = 3$ , then $x^2 = 9$
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 $p \leftrightarrow q$ (if and only if)

Frequently Asked Questions

If the original statement is true, is the converse ever true?

Sometimes. If an angle measures

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.

Why do the converse and the inverse always have the same truth value?

Look at the truth table: the columns for

q \to p

and

\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.

What does if and only if mean?

It means both directions are true:

p \to q

and

q \to p

. We write it

p \leftrightarrow q

. For example,

x + 4 = 10

if and only if

x = 6

How many counterexamples do I need?

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

Formula Card

Original (conditional)

p \to q

If $p$, then $q$

Converse

q \to p

Swap the hypothesis and the conclusion

Inverse

\neg p \to \neg q

Negate both parts, keep the order

Biconditional

p \leftrightarrow q

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

Teacher Guide: Converse and Inverse

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

Shop Promotions

Checking Rules in Computer Programs

Good Definitions

Key Takeaways

Frequently Asked Questions

Glossary

Formula Card

Original (conditional)

Converse

Inverse

Biconditional

More in This Topic

Introduction to Logical Reasoning

Deductive Reasoning

If-Then Statements

Contrapositive