# Logic and Mathematical Statements

## Self-Test:

 1)       Which of the following statements is equivalent to the statement "The cubic root of a rational number is also a rational number"? If $\sqrt[3]{x}$ is a rational number, then $x$ is a rational number. $x$ is a rational number if and only if $\sqrt[3]{x}$ is a rational number. If $x$ is a rational number, then $x^3$ is a rational number. If $x$ is a rational number, then $\sqrt[3]{x}$ is a rational number. None of the Above

Hint 1 Try rewriting the statement in "If... then..." form yourself.
Hint 2 Recall that the cubic root of a number $a$ is a number $b$ so that $b^3 = a$.

 2)       Which of the following statements is true? If there exists a positive integer $x$ such that $x+1<2$, then $x>1$. If there exists a positive integer $x$ such that $x+1<2$, then $x<1$. If there exists a positive integer $x$ such that $x+1<2$, then $x=1$. If there exists a positive integer $x$ such that $x+1<2$, then all real numbers are integers. All of the Above

Hint Try solving the inequality. Don't forget that $x$ must be integer!

 3)       Which of the following statements is false? If $\frac{1}{x}$ is a rational number, then $x$ is a rational number. If $x$ is a non-zero rational number, then $\frac{1}{x}$ is a rational number. If $x$ is a rational number, then $-x$ is a rational number. If $x$ is a rational number, then $\frac{1}{x}$ is a rational number. None of the Above

Hint A rational number is one of the form $\frac{p}{q}$ where, p,q are integers. Use this form to figure out what form $\frac{1}{x}$ and $-x$ must have.

 4)       Which of the following statements is true? If $x$ is a negative integer and $x+3<0$, then $x>1$. If $x$ is a negative integer and $x+3<0$, then $x^{2} <9$. If $x$ is a negative integer and $x+3<0$, then $x^{2} >10$. If $x$ is a negative integer and $x+3<0$, then $x=-3$. None of the above

Hint Try solving the inequalities that appear. Don't forget which way the implication goes.

 5)       Consider the statement "If $x$ is a positive rational number, then $x \leq x^{2}$." If you were trying to determine if this statement is true, you should start by: Assuming that $x$ satisfies $x \leq x^{2}$. Assuming that $x$ is not rational. Assuming that $x$ is a positive rational number. Assuming that $x$ is a positive rational number and that it satisfies $x \leq x^{2}$. None of the above

Hint Don't forget which part of an "If... then..." is the conclusion.

 6)       Which of the following statements is equivalent to the statement "If $x$ is a rational number, then $x+1$ is a rational number."? If $x+1$ is not rational, then $x$ is not rational. If $x$ is not rational, then $x+1$ is not rational. $x$ is rational, but $x+1$ is not rational. $x$ is rational if and only if $x+1$ is rational. None of the Above

Hint The equivalence you're after involves negation.

 7)       The opposite (or negation) of the statement "If $x$ is even, then $\frac{x}{2}$ is odd" is: $x$ is odd and $\frac{x}{2}$ is even. $x$ is even and $\frac{x}{2}$ is even. If $x$ is odd, then $\frac{x}{2}$ is even. If $\frac{x}{2}$ is even, then $x$ is odd. None of the Above

Hint Negation is not just putting "not" in front of everything!

 8)       The opposite (or negation) of the statement "There exists a number $y$, such that for every positive number $x$, $x+y=0$." is: For every $y$ there exists a positive number $x$ so that $x+y \neq 0$. For every $y$ there exists a negative number $x$ so that $x+y = 0$. For every $y$ there exists a negative number $x$ so that $x+y \neq 0$. For every $y$ there does not exist a positive number so that $x+y =0$. None of the Above

Hint Don't forget about the "there exists" and "for every" when negating.

 9)       The statement "$x$ is odd if and only if $\frac{x}{2}$ is not an integer" is equivalent to which pair of statements. "If $x$ is odd, then $\frac{x}{2}$ is an integer" and "If $\frac{x}{2}$ is an integer, then $x$ is odd." "If $x$ is odd, then $\frac{x}{2}$ is not an integer" and "If $\frac{x}{2}$ is not an integer, then $x$ is odd." "If $x$ is not odd, then $\frac{x}{2}$ is not an integer" and "If $\frac{x}{2}$ is not an integer, then $x$ is odd." "If $x$ is even, then $\frac{x}{2}$ is an integer" and "If $x$ is an integer, then $\frac{x}{2}$ is even." None of the Above

Hint Using $\Leftrightarrow$ notation might help. (Think of $\Leftrightarrow$ as being made up of the two arrows $\Rightarrow$ and $\Leftarrow$).

 10)       Consider the statement "If there exists a real number $x$ such that $x^{2} + 1= 0$, then $x<0$." The hypothesis of this statement is: $x<0$. There exists a real number. There exists a real number satisfying $x^{2} + 1 = 0$. There exists a real number such that $x<0$. None of the Above

Hint The hypothesis is the part that you get to assume.