Contrapositive

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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) $\neg p \to q$
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?
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 $10$ .
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 $5$ .
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?
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 $x$ .
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
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.
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?
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.
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
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 $3$ ?
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 $3$ .
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
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 $x = 5$ .
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 $m$ ?
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 $n = 2k + 1$ .
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 $b < 5$ .