Forming a Contrapositive

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.

All Four Related Statements

Start with "If a shape is a square, then it has four right angles." Write the converse, the inverse and the contrapositive, and decide whether each one is true.

Check the original $p \to q$: Every square has four right angles. = True

Converse $q \to p$: If a shape has four right angles, then it is a square. A $3$ by $5$ rectangle has four right angles but is not a square. = False

Inverse $\neg p \to \neg q$: If a shape is not a square, then it does not have four right angles. The same $3$ by $5$ rectangle is a counterexample. = False

Contrapositive $\neg q \to \neg p$: If a shape does not have four right angles, then it is not a square. = True

Answer: The original and the contrapositive are true, the converse and the inverse are false. One counterexample (the rectangle) breaks both the converse and the inverse at the same time, because they are equivalent to each other.

Proving the Equivalence with a Truth Table

Show that $p \to q$ and $\neg q \to \neg p$ have the same truth value in every possible case.

List all combinations of $p$ and $q$: $(T,T)$, $(T,F)$, $(F,T)$, $(F,F)$ = 4 rows

Fill in $p \to q$: An implication is false only when the hypothesis is true and the conclusion is false: row $(T,F)$. = $T, F, T, T$

Fill in $\neg q$ and $\neg p$: Flip each value: $\neg q$ is $F, T, F, T$ and $\neg p$ is $F, F, T, T$. = Two helper columns

Fill in $\neg q \to \neg p$: It is false only where $\neg q$ is $T$ and $\neg p$ is $F$, which happens only in row $(T,F)$. = $T, F, T, T$

Compare the two columns: Both columns read $T, F, T, T$. = Identical in all 4 rows

Answer: The columns match in every row, so $p \to q \equiv \neg q \to \neg p$. The statement and its contrapositive are logically equivalent.

Using the Contrapositive on a Number

The rule "If a number is divisible by $6$, then it is divisible by $3$" is true. Use its contrapositive to decide whether $2750$ is divisible by $6$.

Write the contrapositive: Swap and negate. = If a number is not divisible by $3$, then it is not divisible by $6$.

Test divisibility by $3$ with the digit sum: $2 + 7 + 5 + 0 = 14$, and $14$ is not divisible by $3$. = $2750$ is not divisible by $3$

Check with division: $2750 = 3 \times 916 + 2$ = Remainder $2$

Apply the contrapositive: The hypothesis "not divisible by $3$" is true for $2750$. = $2750$ is not divisible by $6$

Answer: $2750$ is not divisible by $6$. Notice that $2750$ is even, so checking divisibility by $2$ alone would not have settled the question. The contrapositive did.

If n Squared Is Even, Then n Is Even

Explain why "If $n^2$ is even, then $n$ is even" is true for every whole number $n$ by looking at its contrapositive.

Write the contrapositive: Not even means odd for whole numbers. = If $n$ is odd, then $n^2$ is odd.

Write an odd number in general form: $n = 2k + 1$ for some whole number $k$, so $n^2 = (2k + 1)^2 = 4k^2 + 4k + 1$ = $n^2 = 4k^2 + 4k + 1$

Factor out $2$: $n^2 = 2(2k^2 + 2k) + 1$ = Twice a whole number plus $1$, so $n^2$ is odd

Check with $n = 7$: $k = 3$: $7^2 = 49 = 2 \times 24 + 1$ = $49$ is odd

Conclude: The contrapositive is true, and it is equivalent to the original. = The original is true

Answer: Every odd number has an odd square, so the contrapositive is true, and therefore the original statement is true too. The direct route (starting from $n^2$ even) is much harder. Proving a statement this way is called proof by contrapositive, which you will meet in the proofs topic.

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

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

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

Why: 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 $x < 5$.

Mistake: Assuming the converse of a true statement is also true

Why: "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.

Mistake: Getting lost in double negatives

Why: 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."

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.

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 $9$.

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.

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.

Reading League Tables

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

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.

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

A: 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.

Q: What is the difference between the inverse and the contrapositive?

A: 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.

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

A: 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 $6$.

Q: Is a contrapositive the same as a proof by contradiction?

A: 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.