Proof by Contrapositive

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Show your work for each problem.

1.
Which statement is the contrapositive of "If $n^2$ is even, then $n$ is even"?
1. a)If $n^2$ is odd, then $n$ is odd
2. b)If $n$ is odd, then $n^2$ is even
3. c)If $n$ is odd, then $n^2$ is odd
4. d)If $n$ is even, then $n^2$ is even
2.
You want to prove "If $xy$ is odd, then $x$ and $y$ are both odd" by contrapositive. What do you assume at the start?
1. a) $x$ and $y$ are both odd
2. b) $x$ and $y$ are both even
3. c) $x$ is even or $y$ is even
4. d) $xy$ is even
3.
What is the negation of " $x \geq 5$ or $y \geq 5$ "?
1. a) $x \leq 5$ and $y \leq 5$
2. b) $x < 5$ or $y < 5$
3. c) $x < 5$ and $y < 5$
4. d) $x > 5$ and $y > 5$
4.
In a proof by contrapositive of $p \to q$ you start by assuming $\neg q$ . What must you reach at the end?
1. a) $\neg p$
2. b) $p$ and $\neg q$ at the same time
3. c) $q$
4. d)Any contradiction
5.
For which statement is a proof by contrapositive clearly the better choice than a direct proof?
1. a)If $n$ is even, then $n^2$ is even
2. b)If $a$ and $b$ are even, then $a + b$ is even
3. c)If $5n + 1$ is even, then $n$ is odd
4. d)If $n$ is odd, then $n + 1$ is even
6.
In the truth table, look at the row where $p$ is true and $q$ is false. What is the value of $\neg q \to \neg p$ in that row? Type 1 for true and 0 for false.
7.
Contrapositive proof of "if $n^2$ is odd, then $n$ is odd": assume $n = 2k$ , so $n^2 = 4k^2 = 2(2k^2)$ . For $k = 7$ , what is $2k^2$ ?
8.
In the proof that an odd $n$ gives an odd $n^2$ , we write $n = 2k + 1$ and $n^2 = 2(2k^2 + 2k) + 1$ . For $n = 9$ we have $k = 4$ . What is $2k^2 + 2k$ ?
9.
The contrapositive of "if $x + y \geq 10$ , then $x \geq 5$ or $y \geq 5$ " says: if $x < 5$ and $y < 5$ , then $x + y < 10$ . For integers $x$ and $y$ with $x < 5$ and $y < 5$ , what is the largest possible value of $x + y$ ?
10.
To prove "if $n^2 - 6n + 5$ is even, then $n$ is odd" by contrapositive, assume $n = 2k$ . Then $n^2 - 6n + 5 = 2(2k^2 - 6k + 2) + 1$ . For $k = 3$ , what is $2k^2 - 6k + 2$ ?
11.
Check the contrapositive of "if $n^2$ is even, then $n$ is even" with the odd number $n = 11$ .
12.
Prove "if $3n + 2$ is odd, then $n$ is odd" by contrapositive and check it with $k = 4$ .
13.
Contrapositive of "if $xy$ is odd, then $x$ and $y$ are odd", case " $x$ is even". Follow the proof with $x = 6$ and $y = 7$ .
14.
Compare the columns $p \to q$ and $\neg q \to \neg p$ in the truth table with 4 rows.
15.
Prove by contrapositive: for real numbers $x$ , $y$ , $z$ , if $x + y + z \geq 30$ , then $x \geq 10$ or $y \geq 10$ or $z \geq 10$ .
16.
Statement: if $n^2 - 6n + 5$ is even, then $n$ is odd. Check the contrapositive with $n = 2k$ and $k = 5$ .
17.
Prove by contrapositive: if 7 numbers have a sum $S$ greater than 63, then at least one of them is greater than 9.
18.
Prove "if $n^2$ is not divisible by 3, then $n$ is not divisible by 3" by contrapositive and check it with $k = 7$ .