Converse and Inverse

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** $q$.

Forming the Converse

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.

Forming the Inverse

Original: If a number is divisible by 4, then it is even. Write the inverse and test it with the number 6.

Negate the hypothesis: not $p$: a number is not divisible by 4 = $\neg p$

Negate the conclusion: not $q$: it is not even = $\neg q$

Write the inverse: If a number is not divisible by 4, then it is not even. = $\neg p \to \neg q$

Test with 6: $6 \div 4 = 1.5$, so 6 is not divisible by 4, but $6 \div 2 = 3$, so 6 is even. = The inverse is false

Answer: Inverse: If a number is not divisible by 4, then it is not even. It is false: 6 is not divisible by 4, yet 6 is even.

One Counterexample for Both

Original: If $x = 3$, then $x^2 = 9$. Find the truth value of the original, the converse and the inverse.

Check the original: If $x = 3$, then $x^2 = 3 \times 3 = 9$. = Original: true

Write and test the converse: If $x^2 = 9$, then $x = 3$. Try $x = -3$: $(-3)^2 = 9$, but $-3 \neq 3$. = Converse: false

Write and test the inverse: If $x \neq 3$, then $x^2 \neq 9$. Try $x = -3$: $-3 \neq 3$, but $(-3)^2 = 9$. = Inverse: false

Compare: The same number $-3$ breaks the converse and the inverse. = Converse and inverse agree

Answer: The original is true, while the converse and the inverse are both false. The counterexample $x = -3$ works for both.

When Both Directions Are True

Original: If an angle measures $90^\circ$, then it is a right angle. Can you write it as a biconditional?

Check the original: Every angle of $90^\circ$ is a right angle by definition. = Original: true

Write the converse: If an angle is a right angle, then it measures $90^\circ$. = Converse: $q \to p$

Check the converse: Every right angle measures exactly $90^\circ$. There is no counterexample. = Converse: true

Combine both directions: $p \to q$ and $q \to p$ are both true, so $p \leftrightarrow q$. = Biconditional

Answer: Yes: An angle is a right angle if and only if it measures $90^\circ$.

False Original, True Converse

Original: If a number is a multiple of 3, then it is a multiple of 6. Find the truth values of the original, the converse and the inverse.

Test the original: 9 is a multiple of 3, but $9 \div 6 = 1.5$. = Original: false

Test the converse: If a number is a multiple of 6, then it is a multiple of 3. Every multiple of 6 is $6k = 3 \times 2k$. = Converse: true

Test the inverse: If a number is not a multiple of 3, then it is not a multiple of 6. A multiple of 6 would also be a multiple of 3, so this is never broken. = Inverse: true

Draw a conclusion: The truth value of the original tells us nothing about the converse. = Each statement must be checked

Answer: The original is false (counterexample 9), while the converse and the inverse are both true.

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.

Mistake: Assuming the converse is true because the original is true

Why: 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 $x = -3$.

Mistake: Mixing up the converse and the inverse

Why: 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: $\neg p \to \neg q$.

Mistake: Negating only one part when forming the inverse

Why: 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.

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

Why: 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.

Shop Promotions

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

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.

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.

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.

Good Definitions

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

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.

A conditional statement has the form $p \to q$: if $p$, then $q$

The converse swaps the parts: $q \to p$

The inverse negates both parts: $\neg p \to \neg q$

A true statement can have a false converse and a false inverse, for example if $x = 3$, then $x^2 = 9$

The converse and the inverse always have the same truth value, and one counterexample is enough to show a statement is false

When a statement and its converse are both true, they form a biconditional $p \leftrightarrow q$ (if and only if)

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

A: 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.

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

A: 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.

Q: What does if and only if mean?

A: 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 $x = 6$.

Q: How many counterexamples do I need?

A: Just one. A single case where the hypothesis is true and the conclusion is false shows the whole statement is false.

Exit Presentation

Converse and Inverse

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Logic & Sets / Logical Reasoning

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 min

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