Department of Mathematics



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.





Worked Examples and Practice Problems