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
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