Where should you start?

Ten questions, easy to hard. No feedback until the end. Skipping ahead marks earlier lessons as done, and you can still open any of them.

1. Question 1 of 10

   Which of these is a statement?

   • AWhat is $7 + 2$?
   • BClose the door.
   • C$x + 1$
   • D$7$ is an even number.
2. Question 2 of 10

   You know $P \Rightarrow Q$ is true, and you know $P$ is true. What follows?

   • A$Q$ is true
   • B$Q$ is false
   • CNothing follows
   • D$P$ is false
3. Question 3 of 10

   What do you assume at the start of a direct proof of $P \Rightarrow Q$?

   • ANothing
   • B$P$
   • CNot $Q$
   • D$Q$
4. Question 4 of 10

   What is the contrapositive of “If it rains, the street is wet”?

   • AIf the street is wet, it is raining.
   • BIf it rains, the street is dry.
   • CIf the street is not wet, it is not raining.
   • DIf it does not rain, the street is not wet.
5. Question 5 of 10

   To prove “$\sqrt{2}$ is irrational” by contradiction, what do you assume?

   • A$\sqrt{2}$ is greater than $1$
   • B$\sqrt{2}$ is irrational
   • C$\sqrt{2}$ is rational
   • D$\sqrt{2}$ is an integer
6. Question 6 of 10

   How do you prove “for every integer $n$, $n^2 \ge 0$”?

   • ATake an arbitrary integer $n$ and show $n^2 \ge 0$.
   • BCheck $n = 1$, $2$ and $3$, then stop there.
   • CFind one integer for which the claim fails.
   • DFind one integer for which the claim holds.
7. Question 7 of 10

   What are the two parts of a proof by induction?

   • ABase case and inductive step
   • BDefinition and theorem
   • CAssumption and contradiction
   • DExample and counterexample
8. Question 8 of 10

   How does a proof of $A \subseteq B$ begin?

   • ALet $x \notin A$.
   • BAssume $A = B$.
   • CLet $x \in B$.
   • DLet $x \in A$.
9. Question 9 of 10

   Is $f(x) = x^2$ injective as a function $\mathbb{Z} \to \mathbb{Z}$?

   • ANo: $f(1) = f(-1)$ but $1 \ne -1$
   • BYes
   • COnly for positive $x$
   • DOnly when $x$ is even
10. Question 10 of 10

    In the inductive step of strong induction you assume:

    • A$P(1)$, $\ldots$, $P(k)$ all hold
    • B$P(k + 1)$
    • CNothing
    • DOnly $P(k)$