If you know how to prove things by induction, then here is an amazing fact: Theorem. All horses are the same color. Proof. We'll induct on the number of horses. Base case: 1 horse. Clearly with just 1 horse, all horses have the same color. Inductive hypothesis: Assume the claim is true for arbitrary group of N>=1 horses Now, for the inductive step: we'll show that if it is true for any group of N horses, that all have the same color, then it is true for any group of N+1 horses. N+1 horses, you can conclude that the last N horses also have the same color. Therefore all N+1 horses have the same color. QED. What's wrong with this proof? It fails on the induction step when N+1=1 Nothing, all horses are the same color It fails on the induction step when N+1=2 The inductive hypothesis is incorrect Consider the claim: for all xR,nZnx=nx Would you prove it is false by counter-example prove it is true by contradiction prove it is false by contradiction prove it is true directly QUESTION 14 Consider the claim: There exists some integer B that is the largest possible integer. (HINT: If B is an integer, then B+1 is also an integer.) Would you prove it is false by counter-example prove it is true by contradiction prove it is false by contradiction prove it is true directly