“$n$ is even iff $n^2$ is even” and other simple statements to teach proof-writing

I am supposed to teach undergraduate students who do not major in mathematics and I would like to give them a short introduction to mathematical reasoning and to the concept of proof. I am looking for very simple mathematical statements (true or false) to help them get familiar with logic and proof writing.

I am not sure if what I ask is clear, so here are some ideas I came up with:

1. Let $n$ be an integer. If $n$ is a multiple of $42$, then $n$ is even.

2. Let $n$ be an integer. If $n$ is a multiple of $43$, then $n$ is odd.

3. If $x$ is a positive real number, then $x^2 \geq x$.

4. For all $n \in \Bbb Z$, $n$ is even iff $n^2$ is even.

5. There is no $x \in \Bbb Q$ such that $x^2 = 2$.

6. There exist irrational numbers $\alpha,\beta > 0$ such that $\alpha^\beta$ is rational.

7. For all $x \in \Bbb R$, if [for all $\epsilon > 0$, $|x| < \epsilon$], then $x = 0$.

8. There exists $m \in \Bbb Z$ such that, for all $n \in \Bbb Z$, one has $n \leq m$.

9. $(-1)\times (-1) = 1$

10. For all $x \in \Bbb R$, one has $0\times x = 0$. (proposed by Robert Auffarth)

11. Let $n$ be a positive integer. If $x$ is real and $x^n=0$ then $x = 0$. (ibidem)

12. Suppose $b,d \neq 0$ and $\dfrac{a}{b} = \dfrac{c}{d}$. Then $\dfrac{a}{b} + \dfrac{c}{d} = \dfrac{a+c}{b+d}$.

It would be great to cover different type of statements.

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Let $x$ be a real number. Then $x>3$ or $x<6$. (Note the use of inclusive or.) –  Joel Reyes Noche Sep 30 '13 at 13:48
I predict that the students won't like this exercise. If they don't intend to be research mathematicians, they will regard the writing of proofs as pointless formalism. They will say that the propositions are "obvious", and it's a waste of time learning the special language/symbology and structure required to "prove" them the way mathematicians do. –  bubba Oct 3 '13 at 12:42
I'd replace 12 with then $a = c$ and $b = d$. –  Vedran Šego Oct 7 '13 at 14:36
@wilsonw And 2 is true? I see this as a list of simple statements, some true and some not, intended to show how important it is to prove things, and not just have a "hunch". –  Vedran Šego Oct 7 '13 at 14:51
For (9) and (10) I think it is essential to discuss on which basis these would have to be proved. If one does not want to have such discussions, then they may not be good choices. –  Carsten Schultz Oct 7 '13 at 15:15

I left the following exercises for a maths education calculus class --- mainly to test their knowledge of the definitions actually. A few soft questions and maybe more material covered than what you are talking about but many are good proof-writing exercises.

EDIT: I have added some more calculus problems after 36... this time without italics.


1. Suppose that $f:\R\raw\R$ and define a function $g:\R\raw\R$ by $g(x)=1/f(x)$. Prove that $g$ has no roots.

2. Let $n\in\N$ and $f:\R\raw\R$. Prove that $k$ is a root of $[f(x)]^n$ if and only if $k$ is a root of $f$.

3. Let $f:\R\raw\R$. Prove that if $k$ is a root of $f$, then $0$ is a root of $f(x+k)$.

4. Prove that the product of two even functions is an even function.

5. Prove that the composition of two even functions is an even function.

6. Use the unit circle to prove that the cosine function is an even function.

7. Prove that the product of two odd functions is an even function.

8. Suppose that $f:\R\raw\R$ is an odd function defined on the entire real line. Prove that $f$ has a root.

9. Using the fact that sine is an odd function, prove that the tangent function is odd.

10. Prove that $g:\R\raw\R$, defined by $g(x)=-x$ is decreasing.

11. Give an example of a function $f:\R\raw\R$ which is both increasing and decreasing for all $x\in\R$.

12. Give an example of a function that is strictly increasing for all $x\in\R$ but has no roots.

13. Suppose that $f:\R\raw\R$ is strictly increasing on a non-empty closed interval $[a,b]\subset\R$. Show that if $a\leq x_1<x_2\leq b$, then the (secant) line joining $(x_1,f(x_1))$ to $(x_2,f(x_2))$ has positive slope.

14. Suppose that $g:\R\raw\R$ is strictly decreasing and has a root at $a\in\R$. Prove that $g$ has no other roots.

15. Suppose that $f:\R\raw\R$ is a positive increasing function and that $g:\R\raw\R$ is a positive decreasing function. Prove that $q=f/g$, $q(x)=f(x)/g(x)$ is an increasing function.

16. Use the formula for the roots of a quadratic function $p(x)=ax^2+bx+c$ to find an expression for the sum of the roots of $p$; and the product of the roots of $p$.

17. Suppose that $q(x)=ax^2+bx+c$ is a quadratic function with real roots $\alpha$ and $\beta$. Use the fact that quadratic functions are symmetric about the line $x=-b/2a$ --- and that their maxima/ minima are found there to find an expression for $\alpha+\beta$.

18. Let $r(x)=ax^2+bx+c$ be a quadratic function. Use the factor theorem to find an expression for the sum of the roots of $r$; and the product of the roots of $r$.

19. Prove that all polynomials of odd degree have at least one root.

20. Prove the factor theorem for the polynomial $c(x)=ax^3+bx^2+cx+d$.

21. Give an example of degree $4$ polynomials $p$ and $q$ such that $p+q$ is a polynomial of degree $3$.

22. Suppose that $p$ and $q$ are polynomials and let $r$ be the rational function defined by $r(x)=p(x)/q(x)$. Prove that if $k$ is a root of $r$ then $k$ is a root of $p$. By finding a counterexample, show that the converse does not hold.

23. Suppose that $p$ and $q$ are polynomials and let $r$ be the rational function $r(x)=p(x)/q(x)$. If $q(a)=0$, then $r(a)$ is not defined at $a$ and hence discontinuous at $a$. Find examples of polynomials $p$ and $q$ such that: (a) $r$ is continuous, and (b) $r$ is not continuous---but is bounded (there exists a positive number $M>0$ such that $|r(x)|<M$ for all $x\in\R$).

24. Prove that $|x^2+1|=x^2+1$ for all $x\in\R$.

25. Prove that the absolute value function is even.

26. Suppose that $f:\R\raw \R$ has the property that $f(x)<0$ for all $x\in\R$. Describe the relationship between the graph of $f(x)$ and the graph of $|f(x)|$.

27. Let $k\in\R$ be a constant and $a\in\R$. Use the $\varepsilon-\delta$ definition of a limit to prove that $$\lim_{x\raw a}k=k\text{, and }\lim_{x\raw a}x=a.$$

28. Suppose that $f:\R\raw\R$ and $$\lim_{x\raw 1}f(x)=0.$$ Does this imply that $1$ is a root of $f$?

29. Show that there are two values of $a\in\R$ such that the left- and right-handed limits of $f:\R\raw\R$ at $x=1$ agree where: $$f(x)=\left\{\begin{array}{cc}(ax)^2 & \text{ if }x<1 \\ ax+6 & \text{ if }x\geq 1\end{array}\right.$$ Produce a rough sketch of $f$ in each case.

30. Assuming we know what $2^x$ is, we can define a function $f:\R\raw\R$ by: $$f(x)=\frac{1}{1+2^{-1/x}}$$ Sketch an argument that suggests that the left- and right-hand limits of $f(x)$ at $0$ are, respectively, $0$ and $1$.

31. Construct a function $f:\R\raw\R$ such that $$\lim_{x\raw0^-}f(x)=+\infty\text{ , and }\lim_{x\raw 0^+}f(x)=1.$$

32. Suppose that $f:\R\raw\R$ and that $$\forall\varepsilon>0,\,\exists\,\delta>0\text{ such that if }0<|x|<\delta\Rightarrow|f(x)|<\varepsilon.$$ Does this imply that $$\lim_{x\raw 0}f(x)=0.$$

33. For all real numbers $x,\,y\in\R$ with $x\neq y$, there exists a fraction between $x$ and $y$ --- i.e. a $q\in(x,y)$. Consider the function $$f(x)=\left\{\begin{array}{cc}1&\text{ if }x\in\Q \\0 & \text{ if }x\not\in\Q\end{array}\right.$$ Prove that $f$ is not continuous at any point.

34. Consider the function $$g(x)=\left\{\begin{array}{cc}x&\text{ if }x\in\Q \\0 & \text{ if }x\not\in\Q\end{array}\right.$$ Prove that $g$ is continuous at $0$.

35. Suppose that functions $f_1:\R\raw \R$ and $f_2:\R\raw\R$ have the property that, for $i=1,2$ $$f_i(x)=f_i(y)\Raw x=y.$$ Prove that $f=f_1\circ f_2$ has this property also.

36. Find a set $A\subseteq\R$, and functions $f:A\raw\R$ and $g:\R\raw \R$ such that $(g\circ f)(x)=x$ for all $x\in A$ but that there exists a $y\in \R$ such that $(f\circ g)(y)\neq y$.

37. Suppose that $f:[0,1]\raw[0,1]$ is continuous and strictly increasing on $[0,1]$ such that $f(0)=0$ and $f(1)=1$. Suppose further that $g:[0,1]\raw[0,1]$ is a function such that $$(g\circ f)(x)=x$$ for all $x\in[0,1]$. How is the graph of $g$ related to the graph of $f$.

38. Prove that a continuous function is continuous on any non-empty closed interval $[a,b]\subset \R$.

39. Construct a function which is not continuous (everywhere) but is {continuous on} $[0,1]$.

40. The singleton set $\{x\}\subset\R$ is equal to the closed interval $[x,x]$. Verify that all the conclusions of the Intermediate Value Theorem hold when $f:\{x\}\raw\R$ is a continuous function got by restricting a continuous function $f:\R\raw\R$ to $\{x\}$. Find {ALL} such continuous functions $\{x\}\raw\R$.

41. Prove that the Mean Value Theorem holds on any non-empty closed interval for a smooth function $f:\R\raw\R$.

42. Find the local maxima and minima of the constant function $f(x)=0$.

43. Show that $f:\R\raw\R$, $f(x)=x^4+12x^3+54x^2-12x+5$ is concave up on $\R$.

44. Prove that $\tan x$ has a vertical asymptote at $x=\pi/2$.

45. Give an example of a function which is continuous on $(0,1]$ but not on $[0,1]$.

46. Use the Intermediate Value Theorem to prove that $\sqrt{2}\in\R$ exists.

47. Use the Intermediate Value Theorem to prove that if $f:[a,b]\raw\R$ and $g:[a,b]\raw\R$ are continuous functions such that $f(a)=g(b)$ and $f(b)=g(a)$, then there exists a point $c\in[a,b]$ such that $f(c)=g(c)$.

48. It can be shown that a continuous function obtains its absolute maximum and minimum on finite unions of closed intervals. Consider the set $S$ defined by: \begin{align*} S&=[0,2\pi]\cup\left[2\pi,2\pi+\frac{1}{2\pi}\right]\cup\left[3\pi,3\pi+\frac{1}{3\pi}\right]\cup\left[4\pi,4+\frac{1}{4\pi}\right]\cup\cdots\cup\left[100\pi,100\pi+\frac{1}{100\pi}\right], \\ &=[0,2\pi]\cup\left(\bigcup_{i=2}^{100}\left[i\pi,i\pi+\frac{1}{i\pi}\right]\right). \end{align*} Find the absolute maxima and minima of $\sin:S\raw\R$.

49. If $f:(a,b)\raw\R$ is a smooth function such that $f'=0$ for all $x\in(a,b)$ then $f$ is constant on $(a,b)$. Find an example of a differentiable function $g:\R\raw\R$ such that $$\lim_{x\raw\infty}g'(x)=0\text{ , but}\lim_{x\raw\infty}g(x)$$ is not a constant.

50. Sketch continuous functions $f_i:[0,1]\raw\R$ for $i=1,2,3,4,\dots$ such that $f_i$ has $i$ stationary points; i.e. solutions to $f'(c)=0$ for $c\in(0,1)$.

51. Is it possible for a function $f:\R\raw\R$ to have a local maximum/ minimum at a point $a\in\R$ where $f$ is discontinuous?.

52. In the proof of the Closed Interval Method, we make the assumption that $f:[a,b]\raw\R$ is continuous. Is this assumption necessary?.

53. Let $\varepsilon\in (0,1)$ be a constant. Using algebraic techniques, find the critical points of $f:[0,1]\raw\R$ $$f(x):=\left\{\begin{array}{cc}0 & \text{ if }-1\leq x<-\varepsilon \\[1.5ex]\ \frac{1}{\varepsilon}x+1 & \text{ if }-\varepsilon\leq x<0 \\[1.5ex] -\frac{1}{\varepsilon}x+1 & \text{ if }0\leq x<\varepsilon \\[1.5ex] 0 &\text{ if }\varepsilon\leq x\leq 1 \end{array}\right.$$

54. Construct a function $f:\R\raw\R$ that is twice differentiable but not three-times differentiable.

55. Show that $$\lim_{h\raw0}\frac{\cos(\pi+h)-\cos(\pi)}{h}=0.$$ Show that if $f:[a,b]\raw\R$ is twice differentiable, there exists a point $c\in(a,b)$ such that $$f''(c)=\frac{f'(b)-f'(a)}{b-a}$$

56. Suppose that a smooth function $f:[-1,1]\raw\R$ is concave down on $[-1,1]$ and further that $f'(0)=0$. Draw a sketch suggesting that $f$ attains its absolute maximum at $0$.

57. Use the geometric definition of concavity to show that $f(x)=|x|$ is concave up on $\R$.

58. Show that $|x|$ is an asymptotic of $\sqrt{x^2+3}$.

59. Explain why there is no polynomial asymptotic of $\sin x$.

60. Find all vertical asymptotes of $f(x)=\frac{\sin x}{x}$.

61. Use the Intermediate Value Theorem to prove that if $f:[0,1]\rightarrow[0,1]$ is a continuous function, then $f$ has a fixed point; i.e. a point $x\in[0,1]$ such that $f(x)=x$.

62. Suppose that $f:[a,b]\raw \R$ is a differentiable function such that $f'(x)\leq K$ for all $x\in[a,b]$. Prove that $f(a)+K(b-a)$ is an upper bound for the {absolute maximum of $f$ on $[a,b]$} [HINT: Assume that $f(x)>f(a)+K(b-a)$ for some $x\in[a,b]$ and show that this contradicts the Mean Value Theorem.].

63. Call a non-empty set of points $\{x_1,x_2,\dots,x_n\}\subset\R$ an antipodal set for $f:\R\raw\R$ if the tangents to $f$ at $x_1,x_2,\dots,x_n$ are all parallel. Prove that an antipodal sets of a cubic function contain at most two elements.

64. Verify Rolles's Theorem for $\sin:[0,\pi]\raw\R$ geometrically by considering the unit circle with parametric equation $(x,y)=(\cos\theta,\sin\theta)$.

65. Let $[a,b]$ be a non-empty closed interval and consider $\sin:[a,b]\raw\R$. Explain why $\sin$ satisfies the hypothesis of the Mean Value Theorem. Apply the Mean Value Theorem to show that there exists an $x\in(a,b)$ such that $$|\cos(x)|=\left|\frac{\sin(b)-\sin(a)}{b-a}\right|.$$ Now use this result to show that for any $x_1,\,x_2\in \R$ $$|\sin (x_2)-\sin(x_1)|\leq |x_2-x_1|.$$ We could then use this result to prove that $\sin$ is a continuous function. What would be wrong with the proof?

66. In this exercise we see why we might call the quantity $$\frac{f(b)-f(a)}{b-a}$$ the average slope across $[a,b]$ when we talk about the Mean Value Theorem. Let $f:[a,b]\raw\R$ be a differentiable function. Partition the interval $[a,b]$ into $n$ equally spaced points: $$a=:x_0<x_1<x_2<\cdots x_n:=b.$$ If $[a,b]$ is divided into $n$ sub-intervals then each will have width $h=(b-a)/n$ --- so that $x_i=a+ih$. Produce a sketch of a smooth function $f:[a,b]\raw\R$ and a partition of $[a,b]$. Now for large $n$, and hence intervals of small length, smooth curves look like lines so approximate the slope of $f$ on the interval $[x_{i-1},x_{i}]$ by the secant line joining $(x_{i-1},f(x_{i-1}))$ to $(x_{i},f(x_{i}))$ (a sketch should help): $$f'(x)\approx \frac{f(x_i)-f(x_{i-1})}{h}\,\text{ , for }x\in[x_{i-1},x_i].$$ Now average over all of the $n$ sub-intervals: $$\text{average}(f')\approx\frac{\sum_{i=1}^n\left(\frac{f(x_i)-f(x_{i-1})}{h}\right)}{n}.$$ Now use the relationship between $a,\,b,\,n$ and $h$ and telescoping series techniques(see Wikipedia) to show that $$\text{average}(f')\approx \frac{f(b)-f(a)}{b-a}.$$ To make this interpretation precise --- by taking $n\raw\infty$ --- we actually have to use the Mean Value Theorem! More of this in Integration.

67. Argue that if $f:\R\raw\R$ is a smooth, even function such that the only solution to $f'(x)=0$ is $x=0$, then $x=0$ cannot be a saddle point (a point $a\in\R$ such that $f'(a)=0$ but $a$ is not a local maxima or minima). Suppose there is a {local maximum} at $x=0$. Explain why this is an absolute maximum for $f$ on $\R$.

68. Construct a function with vertical asymptotes at $x=0,1,2,3,4,4,\dots,10^6$.

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• There are an infinite number of primes (i.e., the proof by Euclid).

• $a^p\equiv a\mod p$ implies $p$ is prime.

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Let $a,b,c,d$ be positive integers. Prove that if $\frac{a}{b}<\frac{c}{d}$ then $\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}$. (Similarly for $>$ and $=$.)

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An interesting example of a statement they could try to prove is:

Let $k \in \mathbb{Z}$. If $2k$ is odd, then $k$ must be odd.

This seems to be made for contradiction or contrapositive, which work well. However, while it is a true statement, it could be improved. This could lead to a useful discussion about strengthening hypotheses/conclusions, the difference between necessary and sufficient conditions, and vacuous statements. Some of these things may be too advanced for the students you're teaching, but I think the example is simple enough for them to understand if you want to challenge them.

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+1 for a very nice example! –  Joel Reyes Noche Sep 30 '13 at 13:51
I think that it would be a very bad idea to use this as an example in a short introduction: unless a great deal of time is spent on it and the issues that it raises, it is almost certain to generate an enormous amount of confusion. –  Brian M. Scott Oct 1 '13 at 0:09
@BrianM.Scott: I see where you are coming from; I did say that it may be too advanced. I imagine something like this would come after a couple of weeks. I suppose it just depends on the level of the students and how short the introduction is. –  Michael Albanese Oct 1 '13 at 1:05

This may be too hard for non-math students: Any real polynomial of degree $n$ has at most $n$ roots.

Other fun ones that students know but don't know how to prove are:

1. Prove that $0\cdot x=0$ for any real number $x$.

2. Prove that $1\cdot x=x$ for any real number $x$.

3. If $x^n=0$ for a real number $x$, then $x=0$.

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Ok, I put some other ones that my calculus students get mad at! They always say things like "We've known this since high school, why do we have to prove it!" –  Robert Auffarth Sep 30 '13 at 14:00
I like 1 and 3 very much, but I think 2 is more of a definition/axiom (1 is the unit of the ring $(\Bbb R,+,\times)$). Do you expect something involving the construction of the multiplication of real numbers? –  Siméon Oct 1 '13 at 13:22
Yes, you're right, sorry about this. This is usually taken as an axiom! –  Robert Auffarth Oct 1 '13 at 13:27
In the same vein I like "prove $(-1)\times (-1) = 1$". –  Siméon Oct 1 '13 at 13:41
Yes, these simple proofs get students to think in a different way, and they start acquiring nice tricks. –  Robert Auffarth Oct 1 '13 at 13:44

For each $\epsilon >0$, there exists $c$ such that for each $x > c$, $\left| \dfrac{3x-1}{2x+11} -\dfrac{3}{2}\right| < \epsilon$.

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