Contrapositive

Show your work for each problem.

1.
What is the contrapositive of $p \to q$ ?
1. a) $\neg q \to \neg p$
2. b) $q \to p$
3. c) $\neg p \to \neg q$
4. d) $\neg p \to q$
Answer: $\neg q \to \neg p$
The contrapositive swaps and negates: $\neg q \to \neg p$ . The choice $q \to p$ is the converse and $\neg p \to \neg q$ is the inverse.
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?
Answer: 15
$15$ is the only odd number in the list. It is not even, so by the contrapositive it is not a multiple of $4$ . The other three are even, so the contrapositive says nothing about them (in fact $12$ , $20$ and $28$ are all multiples of $4$ ).
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 $10$ .
Answer: 5
1. What is the last digit of $7345$ ? 5
2. The last digit is not $0$ , so by the contrapositive $7345$ is not divisible by $10$ . What is the remainder of $7345 \div 10$ ? 5
4.
Which statement is the contrapositive of "If a number ends in $5$ , then it is divisible by $5$ "?
1. a)If a number is not divisible by $5$ , then it does not end in $5$ .
2. b)If a number is divisible by $5$ , then it ends in $5$ .
3. c)If a number does not end in $5$ , then it is divisible by $5$ .
4. d)If a number does not end in $5$ , then it is not divisible by $5$ .
Answer: If a number is not divisible by $5$ , then it does not end in $5$ .
Swap and negate: "If a number is not divisible by $5$ , then it does not end in $5$ ." The first choice is the converse, the second is the inverse (false: $10$ does not end in $5$ but 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?
Answer: 1
Only the contrapositive is guaranteed, so the answer is $1$ . The converse and the inverse may be true or false; they always agree with each other, not necessarily with the original.
6.
Statement: if $x = 3$ , then $2x = 6$ . Its contrapositive is: if $2x \ne 6$ , then $x \ne 3$ . You know that $2x = 14$ . Show that $x$ is not $3$ and find $x$ .
Answer: 7
1. What is $2 \times 3$ ? 6
2. $14 \ne 6$ , so by the contrapositive $x \ne 3$ . Solve $2x = 14$ : what is $x$ ? 7
7.
A conditional statement $p \to q$ is known to be true. Which related statement must also be true?
1. a)The inverse $\neg p \to \neg q$
2. b)The contrapositive $\neg q \to \neg p$
3. c)The converse $q \to p$
4. d)None of them
Answer: The contrapositive $\neg q \to \neg p$
The contrapositive is logically equivalent to the original, so it must be true. The converse and the inverse can be false, for example for "If a shape is a square, then it has four right angles".
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.
Answer: 0
1. What is the value of $\neg q$ ? 1
2. What is the value of $\neg p$ ? 0
3. What is the value of $\neg q \to \neg p$ ? 0
4. What is the value of $p \to q$ ? 0
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?
Answer: 3
$\neg q \to \neg p$ is false only when $\neg q$ is true and $\neg p$ is false, that is when $p$ is true and $q$ is false. That is $1$ row, so it is true in $3$ rows, exactly like $p \to q$ .
10.
Use $1$ for true and $0$ for false. In the row where $p = 0$ and $q = 1$ , the original $p \to q$ is $1$ . Find the contrapositive and the converse in this row.
Answer: 0
1. What is the value of $\neg q$ ? 0
2. What is the value of $\neg p$ ? 1
3. What is the value of the contrapositive $\neg q \to \neg p$ ? 1
4. What is the value of the converse $q \to p$ ? 0
11.
In the truth table row where $p$ is true and $q$ is false, what are the values of $p \to q$ and $\neg q \to \neg p$ ?
1. a)Both are false
2. b) $p \to q$ is true, $\neg q \to \neg p$ is false
3. c) $p \to q$ is false, $\neg q \to \neg p$ is true
4. d)Both are true
Answer: Both are false
$p \to q$ goes from true to false, so it is false. $\neg q$ is true and $\neg p$ is false, so $\neg q \to \neg p$ also goes from true to false and is false. Both are false.
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 $3$ ?
Answer: 2
$50 = 3 \times 16 + 2$ , so the remainder is $2$ . Since $50$ is not a multiple of $3$ , the contrapositive tells us it is not a multiple of $6$ either.
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 $3$ .
Answer: 2
1. What is the digit sum of $2471$ ? 14
2. What is the remainder when $14$ is divided by $3$ ? 2
3. The digit sum is not divisible by $3$ , so $2471$ is not divisible by $3$ . What is the remainder of $2471 \div 3$ ? 2
14.
Original: "If $x = 4$ , then $x^2 = 16$ ." Which of the four related statements (original, converse, inverse, contrapositive) are true?
1. a)All four
2. b)The original and the contrapositive
3. c)The converse and the inverse
4. d)The original and the converse
Answer: The original and the contrapositive
The original is true, so the contrapositive "If $x^2 \ne 16$ , then $x \ne 4$ " is true. The converse "If $x^2 = 16$ , then $x = 4$ " is false because of $x = -4$ , and so is the inverse "If $x \ne 4$ , then $x^2 \ne 16$ ", with the same counterexample.
15.
The statement "If $3x + 2 > 20$ , then $x > 6$ " is true. Its contrapositive is "If $x \le 6$ , then $3x + 2 \le 20$ ." Check the contrapositive for $x = 6$ and $x = 5$ .
Answer: 17
1. Find the boundary value: solve $3x + 2 = 20$ . What is $x$ ? 6
2. What is $3x + 2$ for $x = 6$ ? 20
3. What is $3x + 2$ for $x = 5$ ? 17
16.
The contrapositive of "If $n^2$ is even, then $n$ is even" is "If $n$ is odd, then $n^2$ is odd." Check it for $n = 13$ : write $13^2$ in the form $2m + 1$ . What is $m$ ?
Answer: 84
$13^2 = 169 = 2 \times 84 + 1$ , so $m = 84$ and $169$ is odd. With $n = 2k + 1$ and $k = 6$ , the general formula gives $m = 2k^2 + 2k = 72 + 12 = 84$ as well.
17.
To see why "If $n^2$ is even, then $n$ is even" holds, check its contrapositive "If $n$ is odd, then $n^2$ is odd" for $n = 9$ , using the form $n = 2k + 1$ .
Answer: 40
1. Write $9 = 2k + 1$ . What is $k$ ? 4
2. What is $n^2$ for $n = 9$ ? 81
3. Write $81 = 2m + 1$ . What is $m$ ? 40
18.
For whole numbers $a, b \ge 0$ : "If $a + b \ge 10$ , then $a \ge 5$ or $b \ge 5$ ." 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 $b < 5$ .
Answer: 8
1. What is the largest whole number $a$ with $a < 5$ ? 4
2. What is the largest whole number $b$ with $b < 5$ ? 4
3. What is the largest possible value of $a + b$ ? 8