Practice Problems

In problem 1, let $A = {1,2,3,4,5}$, $B = {0,2,4,6}$, and $C = {–2,–1,0,1,2,3}$. Which values of $x$ satisfy each statement.

1. a) $x$ is in $A$ and $x$ is in $B$.

b) $x$ is in $A$ or $x$ is in $C$.

c) $x$ is not in $B$ and $x$ is in $C$.

In problems 3 – 5, list or describe all the values of x which make each statement true.

3. a) $x^2 + 3 > 1$

b) $x^3 + 3 > 1$

c) $[ x ] ≤ | x |$

5. a) $x + 5 = 3$ or $x^2 = 9$

b) $x + 5 = 3$ and $x^2 = 9$

c) $| x + 3 | = | x | + 3$

In problem 7, write the contrapositive of each statement. If the statement is false, give a counterexample.

7. a) If $x^2 + x – 6 = 0$ then $x = 2$ or $x = –3$.

b) All triangles have 3 sides.


In problems 9 – 11, write the contrapositive of each statement. If necessary, first write the original statement in the "If . . . then . . . " form.

9. a) If your car is properly tuned, it will get at least 24 miles per gallon.
b) You can have dessert if you eat your vegetables.

11. a) If you love your country, you will vote for me.
b) If guns are outlawed then only outlaws will have guns.

In problems 13 – 15, write the negation of each statement.

13. a) $f(x)$ or $g(x)$ is positive.

b) $x$ is positive.

c) $8$ is a prime number.

15. a) For all numbers $a$ and $b$, $| a + b | = | a | + | b |$.

b) All snakes are poisonous.

c) No dog can climb trees.

17. Write an "If . . . then . . . " statement which is true and whose converse is true.

In problems 19 – 21, state whether each statement is true or false. If the statement is false, give a counterexample.

19. a) If $a$ and $b$ are real numbers then $(a + b)^2 = a^2 + b^2$.

b) If $a > b$ then $a^2 > b^2$.

c) If $a > b$ then $a^3 > b^3$.

21. a) If $f(x)$ and $g(x)$ are linear functions then $f(x) + g(x)$ is a linear function.

b) If $f(x)$ and $g(x)$ are linear functions then $f(x)g(x)$ is a linear function.

c) If $x$ divides 6 then $x$ divides 30.

In problems 23 – 25, rewrite each statement as an "If ... then ... " statement and state whether it is true or false. If the statement is false, give a counterexample.

23. a) The sum of two prime numbers is a prime.

b) The sum of two prime numbers is never a prime.

c) Every prime number is odd. d) Every prime number is even.

25. a) Every solution of $x+5=9$ is odd.

b) Every 3–sided polygon with equal sides is a triangle.

c) Every calculus student studies hard.

d) All (real number) solutions of $x^2 – 5x + 6 = 0$ are even.

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-1.5-Mathematical-Language.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. (a) $x = 2, 4$

(b) $x = –2, –1, 0, 1, 2, 3, 4, 5$

(c) $x = –2, –1, 1, 3$

3. (a) all $x$ (all real numbers)

(b) $x > 3 –2$

(c) all $x$

5. (a) $x = –2, –3, 3$

(b) no values of $x$

(c) $x ≥ 0$

7. (a) If $x ≠ 2$ and $x ≠ –3$, then $x^2 + x – 6 ≠ 0$. True.

(b) If an object does not have 3 sides, then it is not a triangle. True.

9. (a) If your car does not get at least 24 miles per gallon, then it is not tuned properly.

(b) If you can not have dessert, then you did not eat your vegetables.

11. (a) If you will not vote for me, then you do not love your country.

(b) If not only outlaws have guns, then guns are not outlawed. (poor English If someone legally has a gun, then guns are not illegal.

13. (a) Both $f(x)$ and $g(x)$ are not positive.

(b) $x$ is not positive. ( $x ≤ 0$ )

(c) 8 is not a prime number.

15. (a) For some numbers $a$ and $b, | a+b | ≠ | a | + | b |$.

(b) Some snake is not poisonous.

(c) Some dog can climb trees.

17. If $x$ is an integer, then $2x$ is an even integer. True.
Converse: If $2x$ is an even integer, then $x$ is an integer. True.
(It is not likely that these were the statements you thought of. There are lots of other examples).

19. (a) False. Put $a = 3$ and $b = 4$. Then $(a + b)^2 = (7)^2 = 49$, but $a^2 + b^2 = 3^2 + 4^2 = 9 + 16 = 25$.

(b) False. Put $a = –2$ and $b = –3$. Then $a > b$, but $a^2 = 4 < 9 = b^2$.

(c) True.

21. (a) True.

(b) False. Put $f(x) = x + 1$ and $g(x) = x + 2$. Then $f(x)g(x) = x^2 + 3x + 2$ is not a linear function.

(c) True.

23. (a) If $a$ and $b$ are prime numbers, then $a + b$ is prime. False: take $a = 3$ and $b = 5$.

(b) If $a$ and $b$ are prime numbers, then $a + b$ is not prime. False: take $a = 2$ and $b = 3$.

(c) If $x$ is a prime number, then $x$ is odd. False: take $x = 2$. (this is the only counterexample)

(d) If $x$ is a prime number, then $x$ is even. False: take $x = 3$ (or 5 or 7 or ...) 

25. (a) If $x$ is a solution of $x + 5 = 9$, then $x$ is odd. False: take $x = 4$.

(b) If a 3–sided polygon has equal sides, then it is a triangle. True. (We also have nonequilateral triangles .)

(c) If a person is a calculus student, then that person studies hard. False (unfortunately), but we won't mention names.

(d) If $x$ is a (real number) solution of $x^2 – 5x + 6 = 0$, then $x$ is even. False: take $x = 3$.